Catalan numbers: Difference between revisions
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{{Wikipedia}}
{{task|Arithmetic operations}}
<br>
Catalan numbers are a sequence of numbers which can be defined directly:
:<math>C_n = \frac{1}{n+1}{2n\choose n} = \frac{(2n)!}{(n+1)!\,n!} \qquad\mbox{ for }n\ge 0.</math>
Or recursively:
Line 6 ⟶ 10:
:<math>C_0 = 1 \quad \mbox{and} \quad C_n=\frac{2(2n-1)}{n+1}C_{n-1},</math>
;Task:
Implement at least one of these algorithms and print out the first 15 Catalan numbers with each.
[[Memoization]] is not required, but may be worth the effort when using the second method above.
;Related tasks:
*[[Catalan numbers/Pascal's triangle]]
*[[Evaluate binomial coefficients]]
<br><br>
=={{header|11l}}==
<syntaxhighlight lang="11l">V c = 1
L(n) 1..15
print(c)
c = 2 * (2 * n - 1) * c I/ (n + 1)</syntaxhighlight>
{{out}}
<pre>
1
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440
</pre>
=={{header|360 Assembly}}==
Very compact version.
<syntaxhighlight lang="360asm">CATALAN CSECT 08/09/2015
USING CATALAN,R15
LA R7,1 c=1
LA R6,1 i=1
LOOPI CH R6,=H'15' do i=1 to 15
BH ELOOPI
XDECO R6,PG edit i
LR R5,R6 i
SLA R5,1 *2
BCTR R5,0 -1
SLA R5,1 *2
MR R4,R7 *c
LA R6,1(R6) i=i+1
DR R4,R6 /i
LR R7,R5 c=2*(2*i-1)*c/(i+1)
XDECO R7,PG+12 edit c
XPRNT PG,24 print
B LOOPI next i
ELOOPI BR R14
PG DS CL24
YREGS
END CATALAN</syntaxhighlight>
{{out}}
<pre>
1 1
2 2
3 5
4 14
5 42
6 132
7 429
8 1430
9 4862
10 16796
11 58786
12 208012
13 742900
14 2674440
15 9694845
</pre>
=={{header|ABAP}}==
This works for ABAP Version 7.40 and above
<syntaxhighlight lang="abap">
report z_catalan_numbers.
class catalan_numbers definition.
public section.
class-methods:
get_nth_number
importing
i_n type int4
returning
value(r_catalan_number) type int4.
endclass.
class catalan_numbers implementation.
method get_nth_number.
r_catalan_number = cond int4(
when i_n eq 0
then 1
else reduce int4(
init
result = 1
index = 1
for position = 1 while position <= i_n
next
result = result * 2 * ( 2 * index - 1 ) div ( index + 1 )
index = index + 1 ) ).
endmethod.
endclass.
start-of-selection.
do 15 times.
write / |C({ sy-index - 1 }) = { catalan_numbers=>get_nth_number( sy-index - 1 ) }|.
enddo.
</syntaxhighlight>
{{out}}
<pre>
C(0) = 1
C(1) = 1
C(2) = 2
C(3) = 5
C(4) = 14
C(5) = 42
C(6) = 132
C(7) = 429
C(8) = 1430
C(9) = 4862
C(10) = 16796
C(11) = 58786
C(12) = 208012
C(13) = 742900
C(14) = 2674440
</pre>
=={{header|Action!}}==
{{libheader|Action! Tool Kit}}
<syntaxhighlight lang="action!">INCLUDE "D2:REAL.ACT" ;from the Action! Tool Ki
PROC Main()
REAL c,rnom,rden
BYTE n,nom,den
Put(125) PutE() ;clear the screen
IntToReal(1,c)
FOR n=1 TO 15
DO
nom=(n LSH 1-1) LSH 1
den=n+1
IntToReal(nom,rnom)
IntToReal(den,rden)
RealMult(c,rnom,c)
RealDiv(c,rden,c)
PrintF("C(%B)=",n) PrintRE(c)
OD
RETURN</syntaxhighlight>
{{out}}
[https://gitlab.com/amarok8bit/action-rosetta-code/-/raw/master/images/Catalan_numbers.png Screenshot from Atari 8-bit computer]
<pre>
C(1)=1
C(2)=2
C(3)=5
C(4)=14
C(5)=42
C(6)=132
C(7)=429
C(8)=1430
C(9)=4862
C(10)=16796
C(11)=58786
C(12)=208012
C(13)=742900
C(14)=2674440
C(15)=9694845
</pre>
=={{header|Ada}}==
<syntaxhighlight lang="ada">with Ada.Text_IO; use Ada.Text_IO;
procedure Test_Catalan is
Line 25 ⟶ 200:
Put_Line (Integer'Image (N) & " =" & Integer'Image (Catalan (N)));
end loop;
end Test_Catalan;</syntaxhighlight>
{{out|Sample output}}
<pre>
0 = 1
Line 46 ⟶ 220:
15 = 9694845
</pre>
=={{header|ALGOL 68}}==
<syntaxhighlight lang="algol68"># calculate the first few catalan numbers, using LONG INT values #
# (64-bit quantities in Algol 68G which can handle up to C23) #
# returns n!/k! #
PROC factorial over factorial = ( INT n, k )LONG INT:
IF k > n THEN 0
ELIF k = n THEN 1
ELSE # k < n #
LONG INT f := 1;
FOR i FROM k + 1 TO n DO f *:= i OD;
f
FI # factorial over factorial # ;
# returns n! #
PROC factorial = ( INT n )LONG INT:
BEGIN
LONG INT f := 1;
FOR i FROM 2 TO n DO f *:= i OD;
f
END # factorial # ;
# returnss the nth Catalan number using binomial coefficeients #
# uses the factorial over factorial procedure for a slight optimisation #
# note: Cn = 1/(n+1)(2n n) #
# = (2n)!/((n+1)!n!) #
# = factorial over factorial( 2n, n+1 )/n! #
PROC catalan = ( INT n )LONG INT: IF n < 2 THEN 1 ELSE factorial over factorial( n + n, n + 1 ) OVER factorial( n ) FI;
# show the first few catalan numbers #
FOR i FROM 0 TO 15 DO
print( ( whole( i, -2 ), ": ", whole( catalan( i ), 0 ), newline ) )
OD</syntaxhighlight>
{{out}}
<pre>
0: 1
1: 1
2: 2
3: 5
4: 14
5: 42
6: 132
7: 429
8: 1430
9: 4862
10: 16796
11: 58786
12: 208012
13: 742900
14: 2674440
15: 9694845
</pre>
=={{header|ALGOL W}}==
<syntaxhighlight lang="algolw">begin
% print the catalan numbers up to C15 %
integer Cprev;
Cprev := 1; % C0 %
write( s_w := 0, i_w := 3, 0, ": ", i_w := 9, Cprev );
for n := 1 until 15 do begin
Cprev := round( ( ( ( 4 * n ) - 2 ) / ( n + 1 ) ) * Cprev );
write( s_w := 0, i_w := 3, n, ": ", i_w := 9, Cprev );
end for_n
end.</syntaxhighlight>
{{out}}
<pre>
0: 1
1: 1
2: 2
3: 5
4: 14
5: 42
6: 132
7: 429
8: 1430
9: 4862
10: 16796
11: 58786
12: 208012
13: 742900
14: 2674440
15: 9694845
</pre>
=={{header|APL}}==
<syntaxhighlight lang="apl"> {(!2×⍵)÷(!⍵+1)×!⍵}(⍳15)-1</syntaxhighlight>
{{out}}
<pre>1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440</pre>
=={{header|Arturo}}==
<syntaxhighlight lang="rebol">catalan: function [n][
if? n=0 -> 1
else -> div (catalan n-1) * (4*n)-2 n+1
]
loop 0..15 [i][
print [
pad.right to :string i 5
pad.left to :string catalan i 20
]
]</syntaxhighlight>
{{out}}
<pre>0 1
1 1
2 2
3 5
4 14
5 42
6 132
7 429
8 1430
9 4862
10 16796
11 58786
12 208012
13 742900
14 2674440
15 9694845</pre>
=={{header|AutoHotkey}}==
As AutoHotkey has no BigInt, the formula had to be tweaked to prevent overflow. It still fails after n=22
<
out .= "`n" Catalan(A_Index)
Msgbox % clipboard := SubStr(out, 2)
Line 68 ⟶ 358:
Return i
}</
{{out}}
<pre>1
2
Line 85 ⟶ 375:
2674440
9694845</pre>
=={{header|AWK}}==
<syntaxhighlight lang="awk"># syntax: GAWK -f CATALAN_NUMBERS.AWK
BEGIN {
for (i=0; i<=15; i++) {
printf("%2d %10d\n",i,catalan(i))
}
exit(0)
}
function catalan(n, ans) {
if (n == 0) {
ans = 1
}
else {
ans = ((2*(2*n-1))/(n+1))*catalan(n-1)
}
return(ans)
}</syntaxhighlight>
{{out}}
<pre>
0 1
1 1
2 2
3 5
4 14
5 42
6 132
7 429
8 1430
9 4862
10 16796
11 58786
12 208012
13 742900
14 2674440
15 9694845
</pre>
=={{header|BASIC}}==
Line 92 ⟶ 418:
Use of <code>REDIM PRESERVE</code> means this will not work in QBasic (although that could be worked around if desired).
<
REDIM SHARED results(0) AS SINGLE
Line 109 ⟶ 435:
END IF
catalan = results(n)
END FUNCTION</
{{out}}
1 1
2 2
3 5
4 14
5 42
6 132
7 429
8 1430
9 4862
10 16796
11 58786
12 208012
13 742900
14 2674440
15 9694845
==={{header|Applesoft BASIC}}===
{{works with|Chipmunk Basic}}
{{works with|QBasic}}
<syntaxhighlight lang="qbasic">10 HOME : REM 10 CLS for Chipmunk Basic/QBasic
20 DIM c(15)
30 c(0) = 1
40 PRINT 0, c(0)
50 FOR n = 0 TO 14
60 c(n + 1) = 0
70 FOR i = 0 TO n
80 c(n + 1) = c(n + 1) + c(i) * c(n - i)
90 NEXT i
100 PRINT n + 1, c(n + 1)
110 NEXT n
120 END</syntaxhighlight>
==={{header|BASIC256}}===
{{trans|FreeBASIC}}
<syntaxhighlight lang="freebasic">function factorial(n)
if n = 0 then return 1
return n * factorial(n - 1)
end function
function catalan1(n)
prod = 1
for i = n + 2 to 2 * n
prod *= i
next i
return int(prod / factorial(n))
end function
function catalan2(n)
if n = 0 then return 1
sum = 0
for i = 0 to n - 1
sum += catalan2(i) * catalan2(n - 1 - i)
next i
return sum
end function
function catalan3(n)
if n = 0 then return 1
return catalan3(n - 1) * 2 * (2 * n - 1) \ (n + 1)
end function
print "n", "First", "Second", "Third"
print "-", "-----", "------", "-----"
print
for i = 0 to 15
print i, catalan1(i), catalan2(i), catalan3(i)
next i</syntaxhighlight>
{{out}}
<pre>Same as FreeBASIC entry.</pre>
==={{header|BBC BASIC}}===
<syntaxhighlight lang="bbcbasic"> 10 FOR i% = 1 TO 15
20 PRINT FNcatalan(i%)
30 NEXT
40 END
50 DEF FNcatalan(n%)
60 IF n% = 0 THEN = 1
70 = 2 * (2 * n% - 1) * FNcatalan(n% - 1) / (n% + 1)</syntaxhighlight>
{{out}}
<pre> 1
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440</pre>
==={{header|Chipmunk Basic}}===
{{works with|Chipmunk Basic|3.6.4}}
{{trans|Run BASIC}}
<syntaxhighlight lang="qbasic">10 FOR i = 1 TO 15
20 PRINT i;" ";catalan(i)
30 NEXT
40 END
50 SUB catalan(n)
60 catalan = 1
70 IF n <> 0 THEN catalan = ((2*((2*n)-1))/(n+1))*catalan(n-1)
80 END SUB</syntaxhighlight>
==={{header|Craft Basic}}===
<syntaxhighlight lang="basic">dim c[16]
let c[0] = 1
for n = 0 to 15
let p = n + 1
let c[p] = 0
for i = 0 to n
let q = n - i
let c[p] = c[p] + c[i] * c[q]
next i
print n, " ", c[n]
next n</syntaxhighlight>
{{out| Output}}<pre>
0 1
1 1
2 2
3 5
4 14
5 42
6 132
7 429
8 1430
9 4862
10 16796
11 58786
12 208012
13 742900
14 2674440
15 9694845
</pre>
==={{header|FreeBASIC}}===
<syntaxhighlight lang="freebasic">' FB 1.05.0 Win64
Function factorial(n As UInteger) As UInteger
If n = 0 Then Return 1
Return n * factorial(n - 1)
End Function
Function catalan1(n As UInteger) As UInteger
Dim prod As UInteger = 1
For i As UInteger = n + 2 To 2 * n
prod *= i
Next
Return prod / factorial(n)
End Function
Function catalan2(n As UInteger) As UInteger
If n = 0 Then Return 1
Dim sum As UInteger = 0
For i As UInteger = 0 To n - 1
sum += catalan2(i) * catalan2(n - 1 - i)
Next
Return sum
End Function
Function catalan3(n As UInteger) As UInteger
If n = 0 Then Return 1
Return catalan3(n - 1) * 2 * (2 * n - 1) \ (n + 1)
End Function
Print "n", "First", "Second", "Third"
Print "-", "-----", "------", "-----"
Print
For i As UInteger = 0 To 15
Print i, catalan1(i), catalan2(i), catalan3(i)
Next
Print
Print "Press any key to quit"
Sleep</syntaxhighlight>
{{out}}
<pre>
n First Second Third
- ----- ------ -----
0 1 1 1
1 1 1 1
2 2 2 2
3 5 5 5
4 14 14 14
5 42 42 42
6 132 132 132
7 429 429 429
8 1430 1430 1430
9 4862 4862 4862
10 16796 16796 16796
11 58786 58786 58786
12 208012 208012 208012
13 742900 742900 742900
14 2674440 2674440 2674440
15 9694845 9694845 9694845
</pre>
==={{header|FutureBasic}}===
{{trans|FreeBASIC}}
<syntaxhighlight lang="futurebasic">
include "NSLog.incl"
local fn Factorial( n as NSInteger ) as UInt64
UInt64 sum = 0
if n = 0 then sum = 1 : exit fn
sum = n * fn Factorial( n - 1 )
end fn = sum
local fn Catalan1( n as NSInteger ) as UInt64
UInt64 product = 1, result
NSUInteger i
for i = n + 2 to 2 * n
product = product * i
next
result = product / fn Factorial( n )
end fn = result
local fn Catalan2( n as NSInteger ) as UInt64
UInt64 sum = 0
NSUInteger i
if n = 0 then sum = 1 : exit fn
for i = 0 to n - 1
sum += fn Catalan2(i) * fn Catalan2( n - 1 - i )
next
end fn = sum
local fn Catalan3( n as NSInteger ) as UInt64
UInt64 result
if n = 0 then result = 1 : exit fn
result = fn Catalan3( n - 1 ) * 2 * ( 2 * n - 1 ) / ( n + 1 )
end fn = result
NSUInteger i
for i = 0 to 19
if( i < 16 )
NSLog( @"%3d.\t\t%7llu\t\t%12llu\t\t%12llu", i, fn Catalan1( i ), fn Catalan2( i ), fn Catalan3( i ) )
else
NSLog( @"%3d.\t\t%@\t\t%12llu\t\t%12llu", i, @"[-err-]", fn Catalan2( i ), fn Catalan3( i ) )
end if
next
HandleEvents
</syntaxhighlight>
{{output}}
<pre>
0. 1 1 1
1. 1 1 1
2. 2 2 2
3. 5 5 5
4. 14 14 14
5. 42 42 42
6. 132 132 132
7. 429 429 429
8. 1430 1430 1430
9. 4862 4862 4862
10. 16796 16796 16796
11. 58786 58786 58786
12. 208012 208012 208012
13. 742900 742900 742900
14. 2674440 2674440 2674440
15. 9694845 9694845 9694845
16. [-err-] 35357670 35357670
17. [-err-] 129644790 129644790
18. [-err-] 477638700 477638700
19. [-err-] 1767263190 1767263190
</pre>
==={{header|GW-BASIC}}===
{{works with|BASICA}}
<syntaxhighlight lang="gwbasic">
100 REM Catalan numbers
110 DIM C(15)
120 C(0) = 1
130 PRINT 0, C(0)
140 FOR N = 0 TO 14
150 C(N + 1) = 0
160 FOR I = 0 TO N
170 C(N + 1) = C(N + 1) + C(I) * C(N - I)
180 NEXT I
190 PRINT N + 1, C(N + 1)
200 NEXT N
210 END
</syntaxhighlight>
{{out}}
<pre>
0 1
1 1
2 2
Line 127 ⟶ 755:
14 2674440
15 9694845
</pre>
==={{header|Liberty BASIC}}===
{{works with|Just BASIC}}
<syntaxhighlight lang="lb">print "non-recursive version"
print catNonRec(5)
for i = 0 to 15
print i;" = "; catNonRec(i)
next
print
print "recursive version"
print catRec(5)
for i = 0 to 15
print i;" = "; catRec(i)
next
print
print "recursive with memoisation"
redim cats(20) 'clear the array
print catRecMemo(5)
for i = 0 to 15
print i;" = "; catRecMemo(i)
next
print
wait
function catNonRec(n) 'non-recursive version
catNonRec=1
for i=1 to n
catNonRec=((2*((2*i)-1))/(i+1))*catNonRec
next
end function
function catRec(n) 'recursive version
if n=0 then
catRec=1
else
catRec=((2*((2*n)-1))/(n+1))*catRec(n-1)
end if
end function
function catRecMemo(n) 'recursive version with memoisation
if n=0 then
catRecMemo=1
else
if cats(n-1)=0 then 'call it recursively only if not already calculated
prev = catRecMemo(n-1)
else
prev = cats(n-1)
end if
catRecMemo=((2*((2*n)-1))/(n+1))*prev
end if
cats(n) = catRecMemo 'memoisation for future use
end function</syntaxhighlight>
{{out}}
<pre>
non-recursive version
42
0 = 1
1 = 1
2 = 2
3 = 5
4 = 14
5 = 42
6 = 132
7 = 429
8 = 1430
9 = 4862
10 = 16796
11 = 58786
12 = 208012
13 = 742900
14 = 2674440
15 = 9694845
recursive version
42
0 = 1
1 = 1
2 = 2
3 = 5
4 = 14
5 = 42
6 = 132
7 = 429
8 = 1430
9 = 4862
10 = 16796
11 = 58786
12 = 208012
13 = 742900
14 = 2674440
15 = 9694845
recursive with memoisation
42
0 = 1
1 = 1
2 = 2
3 = 5
4 = 14
5 = 42
6 = 132
7 = 429
8 = 1430
9 = 4862
10 = 16796
11 = 58786
12 = 208012
13 = 742900
14 = 2674440
15 = 9694845</pre>
==={{header|Minimal BASIC}}===
{{trans|GW-BASIC}}
<syntaxhighlight lang="gwbasic">
10 REM Catalan numbers
20 DIM C(15)
30 LET C(0) = 1
40 PRINT 0, C(0)
50 FOR N = 0 TO 14
60 LET C(N+1) = 0
70 FOR I = 0 TO N
80 LET C(N+1) = C(N+1)+C(I)*C(N-I)
90 NEXT I
100 PRINT N+1, C(N+1)
110 NEXT N
120 END
</syntaxhighlight>
==={{header|PureBasic}}===
Using the third formula...
<syntaxhighlight lang="purebasic">; saving the division for last ensures we divide the largest
; numerator by the smallest denominator
Procedure.q CatalanNumber(n.q)
If n<0:ProcedureReturn 0:EndIf
If n=0:ProcedureReturn 1:EndIf
ProcedureReturn (2*(2*n-1))*CatalanNumber(n-1)/(n+1)
EndProcedure
ls=25
rs=12
a.s=""
a.s+LSet(RSet("n",rs),ls)+"CatalanNumber(n)"
; cw(a.s)
Debug a.s
For n=0 to 33 ;33 largest correct quad for n
a.s=""
a.s+LSet(RSet(Str(n),rs),ls)+Str(CatalanNumber(n))
; cw(a.s)
Debug a.s
Next</syntaxhighlight>
{{out|Sample Output}}
<pre>
n CatalanNumber(n)
0 1
1 1
2 2
3 5
4 14
5 42
6 132
7 429
8 1430
9 4862
10 16796
11 58786
12 208012
13 742900
14 2674440
15 9694845
16 35357670
17 129644790
18 477638700
19 1767263190
20 6564120420
21 24466267020
22 91482563640
23 343059613650
24 1289904147324
25 4861946401452
26 18367353072152
27 69533550916004
28 263747951750360
29 1002242216651368
30 3814986502092304
31 14544636039226909
32 55534064877048198
33 212336130412243110
</pre>
==={{header|Run BASIC}}===
<syntaxhighlight lang="runbasic">FOR i = 1 TO 15
PRINT i;" ";catalan(i)
NEXT
FUNCTION catalan(n)
catalan = 1
if n <> 0 then catalan = ((2 * ((2 * n) - 1)) / (n + 1)) * catalan(n - 1)
END FUNCTION</syntaxhighlight>
<pre>1 1
2 2
3 5
4 14
5 42
6 132
7 429
8 1430
9 4862
10 16796
11 58786
12 208012
13 742900
14 2674440
15 9694845</pre>
==={{header|Sinclair ZX81 BASIC}}===
Works with 1k of RAM.
The specification asks for the first 15 Catalan numbers. A lot of the other implementations produce either C(0) to C(15), which is 16 numbers, or else C(1) to C(15)—which is 15 numbers, but I'm not convinced they're the first 15. This program produces C(0) to C(14).
<syntaxhighlight lang="basic"> 10 FOR N=0 TO 14
20 LET X=N
30 GOSUB 130
40 LET A=FX
50 LET X=N+1
60 GOSUB 130
70 LET B=FX
80 LET X=2*N
90 GOSUB 130
100 PRINT N,FX/(B*A)
110 NEXT N
120 STOP
130 LET FX=1
140 FOR I=1 TO X
150 LET FX=FX*I
160 NEXT I
170 RETURN</syntaxhighlight>
{{out}}
<pre>0 1
1 1
2 2
3 5
4 14
5 42
6 132
7 429
8 1430
9 4862
10 16796
11 58786
12 208012
13 742900
14 2674440</pre>
==={{header|smart BASIC}}===
<syntaxhighlight lang="qbasic">PRINT "Recursive:"!PRINT
FOR n = 0 TO 15
PRINT n,"#######":catrec(n)
NEXT n
PRINT!PRINT
PRINT "Non-recursive:"!PRINT
FOR n = 0 TO 15
PRINT n,"#######":catnonrec(n)
NEXT n
END
DEF catrec(x)
IF x = 0 THEN
temp = 1
ELSE
n = x
temp = ((2*((2*n)-1))/(n+1))*catrec(n-1)
END IF
catrec = temp
END DEF
DEF catnonrec(x)
temp = 1
FOR n = 1 TO x
temp = (2*((2*n)-1))/(n+1)*temp
NEXT n
catnonrec = temp
END DEF</syntaxhighlight>
==={{header|TI-83 BASIC}}===
This problem is perfectly suited for a TI calculator.
<syntaxhighlight lang="ti-83 basic">:For(I,1,15
:Disp (2I)!/((I+1)!I!
:End</syntaxhighlight>
{{out}}
<pre> 1
2
4
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440
9694845
Done</pre>
==={{Header|Tiny BASIC}}===
Integers are limited to 32767 so only the first ten Catalan numbers can be represented. And even then one has to do some finagling to avoid internal overflows.
{{works with|TinyBasic}}
<syntaxhighlight lang="basic">
10 REM Catalan numbers
20 LET N = 0
30 LET C = 1
40 PRINT N," ",C
50 IF N > 9 THEN END
60 LET N = N + 1
70 GOSUB 200
80 LET C = (2 * N - 1) * C
90 LET C = 2 * C / (N + 1)
100 LET C = C + 2 * I * (2 * N - 1)
110 GOTO 40
200 LET I = 0
210 IF C <= 0 THEN RETURN
220 LET C = C - (N + 1)
230 LET I = I + 1
240 GOTO 210
250 REM To avoid internal overflow, I subtract something clever from
260 REM C and then add it back at the end.</syntaxhighlight>
{{out}}
<pre>0 1
1 1
2 2
3 5
4 14
5 42
6 132
7 429
8 1430
9 4862
10 16796</pre>
==={{header|VBA}}===
<syntaxhighlight lang="vb">Public Sub Catalan1(n As Integer)
'Computes the first n Catalan numbers according to the first recursion given
Dim Cat() As Long
Dim sum As Long
ReDim Cat(n)
Cat(0) = 1
For i = 0 To n - 1
sum = 0
For j = 0 To i
sum = sum + Cat(j) * Cat(i - j)
Next j
Cat(i + 1) = sum
Next i
Debug.Print
For i = 0 To n
Debug.Print i, Cat(i)
Next
End Sub
Public Sub Catalan2(n As Integer)
'Computes the first n Catalan numbers according to the second recursion given
Dim Cat() As Long
ReDim Cat(n)
Cat(0) = 1
For i = 1 To n
Cat(i) = 2 * Cat(i - 1) * (2 * i - 1) / (i + 1)
Next i
Debug.Print
For i = 0 To n
Debug.Print i, Cat(i)
Next
End Sub</syntaxhighlight>
{{out|Result}}
<pre>
Catalan1 15
0 1
1 1
2 2
3 5
4 14
5 42
6 132
7 429
8 1430
9 4862
10 16796
11 58786
12 208012
13 742900
14 2674440
15 9694845
</pre>
(Expect same result with "Catalan2 15")
==={{header|VBScript}}===
<syntaxhighlight lang="vb">
Function catalan(n)
catalan = factorial(2*n)/(factorial(n+1)*factorial(n))
End Function
Function factorial(n)
If n = 0 Then
Factorial = 1
Else
For i = n To 1 Step -1
If i = n Then
factorial = n
Else
factorial = factorial * i
End If
Next
End If
End Function
'Find the first 15 Catalan numbers.
For j = 1 To 15
WScript.StdOut.Write j & " = " & catalan(j)
WScript.StdOut.WriteLine
Next
</syntaxhighlight>
{{Out}}
<pre>
1 = 1
2 = 2
3 = 5
4 = 14
5 = 42
6 = 132
7 = 429
8 = 1430
9 = 4862
10 = 16796
11 = 58786
12 = 208012
13 = 742900
14 = 2674440
15 = 9694845
</pre>
==={{header|Visual Basic .NET}}===
{{trans|C#}}
<syntaxhighlight lang="vbnet">Module Module1
Function Factorial(n As Double) As Double
If n < 1 Then
Return 1
End If
Dim result = 1.0
For i = 1 To n
result = result * i
Next
Return result
End Function
Function FirstOption(n As Double) As Double
Return Factorial(2 * n) / (Factorial(n + 1) * Factorial(n))
End Function
Function SecondOption(n As Double) As Double
If n = 0 Then
Return 1
End If
Dim sum = 0
For i = 0 To n - 1
sum = sum + SecondOption(i) * SecondOption((n - 1) - i)
Next
Return sum
End Function
Function ThirdOption(n As Double) As Double
If n = 0 Then
Return 1
End If
Return ((2 * (2 * n - 1)) / (n + 1)) * ThirdOption(n - 1)
End Function
Sub Main()
Const MaxCatalanNumber = 15
Dim initial As DateTime
Dim final As DateTime
Dim ts As TimeSpan
initial = DateTime.Now
For i = 0 To MaxCatalanNumber
Console.WriteLine("CatalanNumber({0}:{1})", i, FirstOption(i))
Next
final = DateTime.Now
ts = final - initial
Console.WriteLine("It took {0}.{1} to execute", ts.Seconds, ts.Milliseconds)
Console.WriteLine()
initial = DateTime.Now
For i = 0 To MaxCatalanNumber
Console.WriteLine("CatalanNumber({0}:{1})", i, SecondOption(i))
Next
final = DateTime.Now
ts = final - initial
Console.WriteLine("It took {0}.{1} to execute", ts.Seconds, ts.Milliseconds)
Console.WriteLine()
initial = DateTime.Now
For i = 0 To MaxCatalanNumber
Console.WriteLine("CatalanNumber({0}:{1})", i, ThirdOption(i))
Next
final = DateTime.Now
ts = final - initial
Console.WriteLine("It took {0}.{1} to execute", ts.Seconds, ts.Milliseconds)
End Sub
End Module</syntaxhighlight>
{{out}}
<pre>CatalanNumber(0:1)
CatalanNumber(1:1)
CatalanNumber(2:2)
CatalanNumber(3:5)
CatalanNumber(4:14)
CatalanNumber(5:42)
CatalanNumber(6:132)
CatalanNumber(7:429)
CatalanNumber(8:1430)
CatalanNumber(9:4862)
CatalanNumber(10:16796)
CatalanNumber(11:58786)
CatalanNumber(12:208012)
CatalanNumber(13:742900)
CatalanNumber(14:2674440)
CatalanNumber(15:9694845)
It took 0.19 to execute
CatalanNumber(0:1)
CatalanNumber(1:1)
CatalanNumber(2:2)
CatalanNumber(3:5)
CatalanNumber(4:14)
CatalanNumber(5:42)
CatalanNumber(6:132)
CatalanNumber(7:429)
CatalanNumber(8:1430)
CatalanNumber(9:4862)
CatalanNumber(10:16796)
CatalanNumber(11:58786)
CatalanNumber(12:208012)
CatalanNumber(13:742900)
CatalanNumber(14:2674440)
CatalanNumber(15:9694845)
It took 0.831 to execute
CatalanNumber(0:1)
CatalanNumber(1:1)
CatalanNumber(2:2)
CatalanNumber(3:5)
CatalanNumber(4:14)
CatalanNumber(5:42)
CatalanNumber(6:132)
CatalanNumber(7:429)
CatalanNumber(8:1430)
CatalanNumber(9:4862)
CatalanNumber(10:16796)
CatalanNumber(11:58786)
CatalanNumber(12:208012)
CatalanNumber(13:742900)
CatalanNumber(14:2674440)
CatalanNumber(15:9694845)
It took 0.8 to execute</pre>
==={{header|Yabasic}}===
{{trans|FreeBASIC}}
<syntaxhighlight lang="yabasic">print " n First Second Third"
print " - ----- ------ -----"
print
for i = 0 to 15
print i using "###", catalan1(i) using "########", catalan2(i) using "########", catalan3(i) using "########"
next i
end
sub factorial(n)
if n = 0 return 1
return n * factorial(n - 1)
end sub
sub catalan1(n)
local proc, i
prod = 1
for i = n + 2 to 2 * n
prod = prod * i
next i
return int(prod / factorial(n))
end sub
sub catalan2(n)
local sum, i
if n = 0 return 1
sum = 0
for i = 0 to n - 1
sum = sum + catalan2(i) * catalan2(n - 1 - i)
next i
return sum
end sub
sub catalan3(n)
if n = 0 return 1
return ((2 * ((2 * n) - 1)) / (n + 1)) * catalan3(n - 1)
end sub</syntaxhighlight>
{{out}}
<pre>Same as FreeBASIC entry.</pre>
=={{header|Befunge}}==
{{trans|Ada}}
<syntaxhighlight lang="befunge">0>:.:000p1>\:00g-#v_v
v 2-1*2p00 :+1g00\< $
> **00g1+/^v,*84,"="<
_^#<`*53:+1>#,.#+5< @</syntaxhighlight>
{{out}}
<pre>0 = 1
1 = 1
2 = 2
3 = 5
4 = 14
5 = 42
6 = 132
7 = 429
8 = 1430
9 = 4862
10 = 16796
11 = 58786
12 = 208012
13 = 742900
14 = 2674440
15 = 9694845</pre>
=={{header|BQN}}==
<syntaxhighlight lang="bqn">Cat←{ 0⊸<◶⟨1, (𝕊-⟜1)×(¯2+4×⊢)÷1+⊢⟩ 𝕩 }
Fact ← ×´1+↕
Cat1 ← { # direct formula
⌊0.5 + (Fact 2×𝕩) ÷ (Fact 𝕩+1) × Fact 𝕩
}
Cat2 ← { # header based recursion
0: 1;
(𝕊 𝕩-1)×2×(1-˜2×𝕩)÷𝕩+1
}
Cat¨ ↕15
Cat1¨ ↕15
Cat2¨ ↕15</syntaxhighlight>
{{out}}
<pre>⟨ 1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 ⟩</pre>
=={{header|Bracmat}}==
<syntaxhighlight lang="bracmat">( out$straight
& ( C
=
. ( F
= i prod
. !arg:0&1
| 1:?prod
& 0:?i
& whl
' ( 1+!i:~>!arg:?i
& !i*!prod:?prod
)
& !prod
)
& F$(2*!arg)*(F$(!arg+1)*F$!arg)^-1
)
& -1:?n
& whl
' ( 1+!n:~>15:?n
& out$(str$(C !n " = " C$!n))
)
& out$"recursive, with memoization, without fractions"
& :?seenCs
& ( C
= i sum
. !arg:0&1
| ( !seenCs:? (!arg.?sum) ?
| 0:?sum
& -1:?i
& whl
' ( 1+!i:<!arg:?i
& C$!i*C$(-1+!arg+-1*!i)+!sum:?sum
)
& (!arg.!sum) !seenCs:?seenCs
)
& !sum
)
& -1:?n
& whl
' ( 1+!n:~>15:?n
& out$(str$(C !n " = " C$!n))
)
& out$"recursive, without memoization, with fractions"
& ( C
=
. !arg:0&1
| 2*(2*!arg+-1)*(!arg+1)^-1*C$(!arg+-1)
)
& -1:?n
& whl
' ( 1+!n:~>15:?n
& out$(str$(C !n " = " C$!n))
)
& out$"Using taylor expansion of sqrt(1-4X). (See http://bababadalgharaghtakamminarronnkonnbro.blogspot.in/2012/10/algebraic-type-systems-combinatorial.html)"
& out$(1+(1+-1*tay$((1+-4*X)^1/2,X,16))*(2*X)^-1+-1)
& out$
);</syntaxhighlight>
{{out}}
<syntaxhighlight lang="bracmat">straight
C0 = 1
C1 = 1
C2 = 2
C3 = 5
C4 = 14
C5 = 42
C6 = 132
C7 = 429
C8 = 1430
C9 = 4862
C10 = 16796
C11 = 58786
C12 = 208012
C13 = 742900
C14 = 2674440
C15 = 9694845
recursive, with memoization, without fractions
C0 = 1
C1 = 1
C2 = 2
C3 = 5
C4 = 14
C5 = 42
C6 = 132
C7 = 429
C8 = 1430
C9 = 4862
C10 = 16796
C11 = 58786
C12 = 208012
C13 = 742900
C14 = 2674440
C15 = 9694845
recursive, without memoization, with fractions
C0 = 1
C1 = 1
C2 = 2
C3 = 5
C4 = 14
C5 = 42
C6 = 132
C7 = 429
C8 = 1430
C9 = 4862
C10 = 16796
C11 = 58786
C12 = 208012
C13 = 742900
C14 = 2674440
C15 = 9694845
Using taylor expansion of sqrt(1-4X). (See http://bababadalgharaghtakamminarronnkonnbro.blogspot.in/2012/10/algebraic-type-systems-combinatorial.html)
1
+ X
+ 2*X^2
+ 5*X^3
+ 14*X^4
+ 42*X^5
+ 132*X^6
+ 429*X^7
+ 1430*X^8
+ 4862*X^9
+ 16796*X^10
+ 58786*X^11
+ 208012*X^12
+ 742900*X^13
+ 2674440*X^14
+ 9694845*X^15
</syntaxhighlight>
=={{header|Brat}}==
<
true? n == 0
{ 1 }
Line 137 ⟶ 1,561:
0.to 15 { n |
p "#{n} - #{catalan n}"
}</
{{out}}
<pre>0 - 1
1 - 1
Line 156 ⟶ 1,580:
15 - 9694845
</pre>
=={{header|C}}==
All three methods mentioned in the task:
<syntaxhighlight lang="c">#include <stdio.h>
typedef unsigned long long ull;
ull binomial(ull m, ull n)
{
ull r = 1, d = m - n;
if (d > n) { n = d; d = m - n; }
while (m > n) {
r *= m--;
while (d > 1 && ! (r%d) ) r /= d--;
}
return r;
}
ull catalan1(int n) {
return binomial(2 * n, n) / (1 + n);
}
ull catalan2(int n) {
int i;
ull r = !n;
for (i = 0; i < n; i++)
r += catalan2(i) * catalan2(n - 1 - i);
return r;
}
ull catalan3(int n)
{
return n ? 2 * (2 * n - 1) * catalan3(n - 1) / (1 + n) : 1;
}
int main(void)
{
int i;
puts("\tdirect\tsumming\tfrac");
for (i = 0; i < 16; i++) {
printf("%d\t%llu\t%llu\t%llu\n", i,
catalan1(i), catalan2(i), catalan3(i));
}
return 0;
}</syntaxhighlight>
{{out}}
direct summing frac
0 1 1 1
1 1 1 1
2 2 2 2
3 5 5 5
4 14 14 14
5 42 42 42
6 132 132 132
7 429 429 429
8 1430 1430 1430
9 4862 4862 4862
10 16796 16796 16796
11 58786 58786 58786
12 208012 208012 208012
13 742900 742900 742900
14 2674440 2674440 2674440
15 9694845 9694845 9694845
=={{header|C sharp|C#}}==
<syntaxhighlight lang="csharp">namespace CatalanNumbers
{
/// <summary>
Line 272 ⟶ 1,760:
}
}
}</syntaxhighlight>
{{out}}
<pre>
CatalanNumber(0):1
Line 330 ⟶ 1,817:
It took 0.3 to execute
</pre>
=={{header|C++}}==
===4 Classes===
We declare 4 classes representing the four different algorithms for calculating Catalan numbers as given in the description of the task. In addition, we declare two supporting classes for the calculation of factorials and binomial coefficients. Because these two are only internal supporting code they are hidden in namespace 'detail'. Overloading the function call operator to execute the calculation is an obvious decision when using C++. (algorithms.h)
<
#define __ALGORITHMS_H__
Line 586 ⟶ 1,879:
} //namespace rosetta
#endif //!defined __ALGORITHMS_H__</
Here is the implementation of the algorithms. The c'tor of each class tells us the algorithm which will be used. (algorithms.cpp)
<
using std::cout;
using std::endl;
Line 688 ⟶ 1,981:
product *= (double(n - (k - i)) / i);
return (unsigned long long)(floor(product + 0.5));
}</
In order to test what we have done, a class Test is created. Using the template parameters N (number of Catalan numbers to be calculated) and A (the kind of algorithm to be used) the compiler will create code for all the test cases we need. What would C++ be without templates ;-) (tester.h)
<
#define __TESTER_H__
Line 715 ⟶ 2,008:
} //namespace rosetta
#endif //!defined __TESTER_H__</
Finally, we test the four different algorithms. Note that the first one (direct calculation using the factorial) only works up to N = 10 because some intermediate result (namely (2n)! with n = 11) exceeds the boundaries of an unsigned 64 bit integer. (catalanNumbersTest.cpp)
<
#include "tester.h"
using namespace rosetta::catalanNumbers;
Line 728 ⟶ 2,021:
Test<15, CatalanNumbersRecursiveSum>::Do();
return 0;
}</
{{out}}
(source code is compiled both by MS Visual C++ 10.0 (WinXP 32 bit) and GNU g++ 4.4.3 (Ubuntu 10.04 64 bit) compilers)
<pre>
Direct calculation using the factorial
Line 795 ⟶ 2,089:
C(15) = 9694845
</pre>
=={{header|Clojure}}==
<
(defn catalan-numbers-direct []
Line 813 ⟶ 2,106:
user> (take 15 (catalan-numbers-recursive))
(1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440)</
=={{header|CLU}}==
<syntaxhighlight lang="clu">catalan = iter (amount: int) yields (int)
c: int := 1
for n: int in int$from_to(1, amount) do
yield(c)
c := (4*n-2)*c/(n+1)
end
end catalan
start_up = proc ()
po: stream := stream$primary_output()
for n: int in catalan(15) do
stream$putl(po, int$unparse(n))
end
end start_up</syntaxhighlight>
{{out}}
<pre>1
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440</pre>
=={{header|Common Lisp}}==
With all three methods defined.
<
;; factorial. CLISP actually has "!" defined for this
(labels ((! (x) (if (zerop x) 1 (* x (! (1- x))))))
Line 844 ⟶ 2,168:
for i from 1 to 3 do
(format t "~%Method ~d:~%" i)
(dotimes (i 16) (format t "C(~2d) = ~d~%" i (
=={{header|Cowgol}}==
<syntaxhighlight lang="cowgol">include "cowgol.coh";
sub catalan(n: uint32): (c: uint32) is
c := 1;
var i: uint32 := 1;
while i <= n loop
c := (4*i-2)*c/(i+1);
i := i+1;
end loop;
end sub;
var i: uint8 := 0;
while i < 15 loop
print("catalan(");
print_i8(i);
print(") = ");
print_i32(catalan(i as uint32));
print_nl();
i := i+1;
end loop;</syntaxhighlight>
{{out}}
<pre>catalan(0) = 1
catalan(1) = 1
catalan(2) = 2
catalan(3) = 5
catalan(4) = 14
catalan(5) = 42
catalan(6) = 132
catalan(7) = 429
catalan(8) = 1430
catalan(9) = 4862
catalan(10) = 16796
catalan(11) = 58786
catalan(12) = 208012
catalan(13) = 742900
catalan(14) = 2674440</pre>
=={{header|Crystal}}==
{{trans|Ruby}}
<syntaxhighlight lang="ruby">require "big"
require "benchmark"
def factorial(n : BigInt) : BigInt
(1..n).product(1.to_big_i)
end
def factorial(n : Int32 | Int64)
factorial n.to_big_i
end
# direct
def catalan_direct(n)
factorial(2*n) / (factorial(n + 1) * factorial(n))
end
# recursive
def catalan_rec1(n)
return 1 if n == 0
(0...n).reduce(0) do |sum, i|
sum + catalan_rec1(i) * catalan_rec1(n - 1 - i)
end
end
def catalan_rec2(n)
return 1 if n == 0
2*(2*n - 1) * catalan_rec2(n - 1) / (n + 1)
end
# performance and results
Benchmark.bm do |b|
b.report("catalan_direct") { 16.times { |n| catalan_direct(n) } }
b.report("catalan_rec1") { 16.times { |n| catalan_rec1(n) } }
b.report("catalan_rec2") { 16.times { |n| catalan_rec2(n) } }
end
puts "\n direct rec1 rec2"
16.times { |n| puts "%2d :%9d%9d%9d" % [n, catalan_direct(n), catalan_rec1(n), catalan_rec2(n)] }
</syntaxhighlight>
{{out}}
<pre>
# with --release flag
# Using: i7-6700HQ, 3.5GHz
user system total real
catalan_direct 0.000026 0.000052 0.000078 ( 0.000074)
catalan_rec1 0.139766 0.001143 0.140909 ( 0.141418)
catalan_rec2 0.000003 0.000000 0.000003 ( 0.000003)
direct rec1 rec2
0 : 1 1 1
1 : 1 1 1
2 : 2 2 2
3 : 5 5 5
4 : 14 14 14
5 : 42 42 42
6 : 132 132 132
7 : 429 429 429
8 : 1430 1430 1430
9 : 4862 4862 4862
10 : 16796 16796 16796
11 : 58786 58786 58786
12 : 208012 208012 208012
13 : 742900 742900 742900
14 : 2674440 2674440 2674440
15 : 9694845 9694845 9694845
</pre>
=={{header|D}}==
<
auto product(R)(R r) { return reduce!q{a * b}(1.BigInt, r); }
const cats1 = sequence!((a, n) => iota(n+2, 2*n+1).product / iota(1, n+1).product)(1);
BigInt
alias mcats2a = memoize!
if (n == 0) return
return n.iota.map!(i => mcats2a(i) * mcats2a(n - 1 - i)).sum;
}
const cats2 = sequence!((a, n) => n.cats2a);
const cats3 =
void main() {
foreach (
cats.take(15).writeln;
}</
{{out}}
<pre>[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440]
[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440]
[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440]</pre>
=={{header|Delphi}}==
See [https://www.rosettacode.org/wiki/Catalan_numbers#Pascal Pascal].
=={{header|EasyLang}}==
<syntaxhighlight lang="text">
func catalan n .
if n = 0
return 1
.
return 2 * (2 * n - 1) * catalan (n - 1) div (1 + n)
.
for i = 0 to 14
print catalan i
.
</syntaxhighlight>
=={{header|
{{incorrect|Echolisp|series starts 1, 1, 2, ...}}
<syntaxhighlight lang="scheme">
(lib 'sequences)
(lib 'bigint)
(lib 'math)
;; function definition
(define (C1 n) (/ (factorial (* n 2)) (factorial (1+ n)) (factorial n)))
(for ((i [1 .. 16])) (write (C1 i)))
→ 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845
;; using a recursive procedure with memoization
(define (C2 n) ;; ( Σ ...)is the same as (sigma ..)
(Σ (lambda(i) (* (C2 i) (C2 (- n i 1)))) 0 (1- n)))
(remember 'C2 #(1)) ;; first term defined here
(for ((i [1 .. 16])) (write (C2 i)))
→ 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845
;; using procrastinators = infinite sequence
(define (catalan n acc) (/ (* acc 2 (1- (* 2 n))) (1+ n)))
(define C3 (scanl catalan 1 [1 ..]))
(take C3 15)
→ (1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845)
;; the same, using infix notation
(lib 'match)
(load 'infix.glisp)
(define (catalan n acc) ((2 * acc * ( 2 * n - 1)) / (n + 1)))
(define C3 (scanl catalan 1 [1 ..]))
(take C3 15)
→ (1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845)
;; or
(for ((c C3) (i 15)) (write c))
→ 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845
</syntaxhighlight>
=={{header|EDSAC order code}}==
The Catalan numbers are here computed by the second method, as a sum of products.
The program illustrates a difficulty with multiplying integers on the EDSAC,
which was designed to work with fixed-point numbers in the range [-1, 1).
When we store a 35-bit integer A, we are really storing A*(2^-34).
As long as integer arithmetic is confined to addition and subtraction, this scaling can be ignored.
But if we multiply two 35-bit integers A and B, the result is really A*B*(2^-68),
and needs to be multiplied by 2^34 to get the same scaling as for A and B.
<syntaxhighlight lang="edsac">
[Calculation of Catalan numbers.
EDSAC program, Initial Orders 2.]
[Define where to store the list of Catalan numbers.]
T 54 K [store address in location 54, so that values
are accessed by code letter C (for Catalan)]
P 200 F [<------ address here]
[Modification of library subroutine P7.
Prints signed integer up to 10 digits, right-justified.
54 storage locations; working position 4D.
Must be loaded at an even address.
Input: Number is at 0D.]
T 56 K
GKA3FT42@A47@T31@ADE10@T31@A48@T31@SDTDH44#@NDYFLDT4DS43@TF
H17@S17@A43@G23@UFS43@T1FV4DAFG50@SFLDUFXFOFFFSFL4FT4DA49@T31@
A1FA43@G20@XFP1024FP610D@524D!FO46@O26@XFO46@SFL8FT4DE39@
[Main routine]
T 120 K [load at 120]
G K [set @ (theta) to load address]
[Variables]
[0] P F [index of Catalan number]
[Constants]
[1] P 7 D [maximum index required]
[2] P D [single-word 1]
[3] P 2 F [to change addresses by 2]
[4] H #C [these 3 are used to manufacture EDSAC orders]
[5] T #C
[6] V #C
[7] K 4096 F [(1) add to change T order into H order
(2) teleprinter null]
[8] # F [figures shift]
[9] ! F [space]
[10] @ F [carriage return]
[11] & F [line feed]
[Enter with acc = 0]
[12] O 8 @ [set teleprinter to figures]
T 4 D [clear 5F and sandwich bit]
A 2 @ [load single-word 1]
T 4 F [store as double word at 4D; clear acc]
[Here with index in acc, Catalan number in 4D]
[16] U @ [store index]
L 1 F [times 4 by shifting]
A 5 @ [make T order to store Catalan number]
U 27 @ [plant in code]
A 7 @ [make H order with same address]
U 45 @ [plant in code]
S 47 @ [make A order with same address]
T 34 @ [plant in code]
A 6 @ [load V order for start of list]
T 46 @ [plant in code]
A 4 D [Catalan number from temp store]
[27] T #C [store in list (manufactured order)]
T D [clear 1F and sandwich bit]
A @ [load single-word index]
T F [store as double word at 0D]
[31] A 31 @ [for return from print subroutine]
G 56 F [print index]
O 9 @ [followed by space]
[34] A #C [load Catalan number (manufactured order)]
T D [to 0D for printing]
[36] A 36 @ [for return from print subroutine]
G 56 F [print Catalan number]
O 10 @ [followed by new line]
O 11 @
T 4 D [clear partial sum]
A @ [load index]
S 1 @ [reached the maximum?]
E 64 @ [if so, jump to exit]
[Inner loop to compute sum of products C{i}*C(n-1}]
[44] T F [clear acc]
[45] H #C [C{n-i} to mult reg (manufactured order)]
[46] V #C [acc := C{i}*C{n-i} (manufactiured order)]
[Multiply product by 2^34 (see preamble). The 'L F' order is
also exploited above to convert an H order into an A order.]
[47] L F [shift acc left by 13 (the maximum available)]
L F [shift 13 more]
L 64 F [shift 8 more, total 34]
A 4 D [add partial sum]
T 4 D [update partial sum]
A 46 @ [inc i in V order]
A 3 @
T 46 @
A 45 @ [dec (n - i) in H order]
S 3 @
U 45 @
S 4 @ [is (n - i) now negative?]
E 44 @ [if not, loop back]
[Here with latest Catalan number in temp store 4D]
T F [clear acc]
A @ [load index]
A 2 @ [add 1]
E 16 @ [back to start of outer loop]
[64] O 7 @ [exit; print null to flush teleprinter buffer]
Z F [stop]
E 12 Z [define entry point]
P F [acc = 0 on entry]
</syntaxhighlight>
{{out}}
<pre>
0 1
1 1
2 2
3 5
4 14
5 42
6 132
7 429
8 1430
9 4862
10 16796
11 58786
12 208012
13 742900
14 2674440
15 9694845
</pre>
=={{header|Eiffel}}==
<syntaxhighlight lang="eiffel">
class
APPLICATION
create
make
feature {NONE}
make
do
across
0 |..| 14 as c
loop
io.put_double (nth_catalan_number (c.item))
io.new_line
end
end
nth_catalan_number (n: INTEGER): DOUBLE
--'n'th number in the sequence of Catalan numbers.
require
n_not_negative: n >= 0
local
s, t: DOUBLE
do
if n = 0 then
Result := 1.0
else
t := 4 * n.to_double - 2
s := n.to_double + 1
Result := t / s * nth_catalan_number (n - 1)
end
end
end
</syntaxhighlight>
{{out}}
<pre>
1
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440
</pre>
=={{header|Elixir}}==
{{trans|Erlang}}
<syntaxhighlight lang="elixir">defmodule Catalan do
def cat(n), do: div( factorial(2*n), factorial(n+1) * factorial(n) )
defp factorial(n), do: fac1(n,1)
defp fac1(0, acc), do: acc
defp fac1(n, acc), do: fac1(n-1, n*acc)
def cat_r1(0), do: 1
def cat_r1(n), do: Enum.sum(for i <- 0..n-1, do: cat_r1(i) * cat_r1(n-1-i))
def cat_r2(0), do: 1
def cat_r2(n), do: div(cat_r2(n-1) * 2 * (2*n - 1), n + 1)
def test do
range = 0..14
:io.format "Directly:~n~p~n", [(for n <- range, do: cat(n))]
:io.format "1st recusive method:~n~p~n", [(for n <- range, do: cat_r1(n))]
:io.format "2nd recusive method:~n~p~n", [(for n <- range, do: cat_r2(n))]
end
end
Catalan.test</syntaxhighlight>
{{out}}
<pre>
Directly:
[1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440]
1st recusive method:
[1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440]
2nd recusive method:
[1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440]
</pre>
=={{header|Erlang}}==
<syntaxhighlight lang="erlang">-module(catalan).
-export([test/0]).
cat(N) ->
factorial(2 * N) div (factorial(N+1) * factorial(N)).
factorial(N) ->
fac1(N,1).
fac1(0,Acc) ->
Acc;
fac1(N,Acc) ->
fac1(N-1, N * Acc).
cat_r1(0) ->
1;
cat_r1(N) ->
lists:sum([cat_r1(I)*cat_r1(N-1-I) || I <- lists:seq(0,N-1)]).
cat_r2(0) ->
1;
cat_r2(N) ->
cat_r2(N - 1) * (2 * ((2 * N) - 1)) div (N + 1).
test() ->
TestList = lists:seq(0,14),
io:format("Directly:\n~p\n",[[cat(N) || N <- TestList]]),
io:format("1st recusive method:\n~p\n",[[cat_r1(N) || N <- TestList]]),
io:format("2nd recusive method:\n~p\n",[[cat_r2(N) || N <- TestList]]).</syntaxhighlight>
{{out}}
<pre>
Directly:
[1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440]
1st recusive method:
[1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440]
2nd recusive method:
[1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440]
</pre>
=={{header|ERRE}}==
<syntaxhighlight lang="text">PROGRAM CATALAN
PROCEDURE CATALAN(N->RES)
RES=1
FOR I=1 TO N DO
RES=RES*2*(2*I-1)/(I+1)
END FOR
END PROCEDURE
BEGIN
FOR N=0 TO 15 DO
CATALAN(N->RES)
PRINT(N;"=";RES)
END FOR
END PROGRAM
</syntaxhighlight>
{{out}}
<pre>
0 = 1
1 = 1
2 = 2
3 = 5
4 = 14
5 = 42
6 = 132
7 = 429
8 = 1430
9 = 4862
10 = 16796
11 = 58786
12 = 208012
13 = 742900
14 = 2674440
15 = 9694845
</pre>
=={{header|Euphoria}}==
<syntaxhighlight lang="euphoria">--Catalan number task from Rosetta Code wiki
--User:Lnettnay
--function from factorial task
function factorial(integer n)
atom f = 1
while n > 1 do
f *= n
n -= 1
end while
return f
end function
function catalan(integer n)
atom numerator = factorial(2 * n)
atom denominator = factorial(n+1)*factorial(n)
return numerator/denominator
end function
for i = 0 to 15 do
? catalan(i)
end for</syntaxhighlight>
{{out}}
<pre>
1
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440
9694845
</pre>
=={{header|F_Sharp|F#}}==
<syntaxhighlight lang="fsharp">
Seq.unfold(fun (c,n) -> let cc = 2*(2*n-1)*c/(n+1) in Some(c,(cc,n+1))) (1,1) |> Seq.take 15 |> Seq.iter (printf "%i, ")
</syntaxhighlight>
{{out}}
<pre>
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440,
</pre>
=={{header|Factor}}==
The first method:
<syntaxhighlight lang="factor">USING: kernel math math.combinatorics prettyprint ;
: catalan ( n -- n ) [ 1 + recip ] [ 2 * ] [ nCk * ] tri ;
15 [ catalan . ] each-integer</syntaxhighlight>
{{out}}
<pre>
1
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440
</pre>
The last method, memoized by using arrays.
<syntaxhighlight lang="factor">USING: kernel math prettyprint sequences ;
: next ( seq -- newseq )
[ ] [ last ] [ length ] tri
[ 2 * 1 - 2 * ] [ 1 + ] bi /
* suffix ;
: Catalan ( n -- seq ) V{ 1 } swap 1 - [ next ] times ;
15 Catalan .</syntaxhighlight>
{{out}}
Similar to above.
=={{header|Fantom}}==
{{incorrect|Fantom|series starts 1, 1, 2, ...}}
<syntaxhighlight lang="fantom">class Main
{
static Int factorial (Int n)
Line 946 ⟶ 2,802:
}
}
}</syntaxhighlight>
22! exceeds the range of Fantom's Int class, so the first technique fails afer n=10
<pre>
1 1 1 1
Line 968 ⟶ 2,821:
15 -2 9694845 9694845
</pre>
=={{header|Fermat}}==
<syntaxhighlight lang="fermat">Func Catalan(n)=(2*n)!/((n+1)!*n!).;
for i=1 to 15 do !Catalan(i);!' ' od;</syntaxhighlight>
{{out}}<pre>
1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845
</pre>
=={{header|Forth}}==
<
=={{header|Fortran}}==
{{works with|Fortran|90 and later}}
<
!=======================================================================================
implicit none
Line 1,018 ⟶ 2,874:
!=======================================================================================
end function catalan_numbers</syntaxhighlight>
{{out}}
<pre>
===============
Line 1,046 ⟶ 2,899:
=={{header|Frink}}==
Frink includes efficient algorithms for calculating arbitrarily-large binomial coefficients and automatically caches factorials.
<syntaxhighlight lang="frink">catalan[n] := binomial[2n,n]/(n+1)
for n = 0 to 15
println[catalan[n]]</syntaxhighlight>
=={{header|FunL}}==
<syntaxhighlight lang="funl">import integers.choose
import util.TextTable
def
catalan( n ) = choose( 2n, n )/(n + 1)
catalan2( n ) = product( (n + k)/k | k <- 2..n )
catalan3( 0 ) = 1
catalan3( n ) = 2*(2n - 1)/(n + 1)*catalan3( n - 1 )
t = TextTable()
t.header( 'n', 'definition', 'product', 'recursive' )
t.line()
for i <- 1..4
t.rightAlignment( i )
for i <- 0..15
t.row( i, catalan(i), catalan2(i), catalan3(i) )
println( t )</syntaxhighlight>
{{out}}
<pre>
+----+------------+---------+-----------+
| n | definition | product | recursive |
+----+------------+---------+-----------+
| 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 |
| 2 | 2 | 2 | 2 |
| 3 | 5 | 5 | 5 |
| 4 | 14 | 14 | 14 |
| 5 | 42 | 42 | 42 |
| 6 | 132 | 132 | 132 |
| 7 | 429 | 429 | 429 |
| 8 | 1430 | 1430 | 1430 |
| 9 | 4862 | 4862 | 4862 |
| 10 | 16796 | 16796 | 16796 |
| 11 | 58786 | 58786 | 58786 |
| 12 | 208012 | 208012 | 208012 |
| 13 | 742900 | 742900 | 742900 |
| 14 | 2674440 | 2674440 | 2674440 |
| 15 | 9694845 | 9694845 | 9694845 |
+----+------------+---------+-----------+
</pre>
=={{header|Fōrmulæ}}==
{{FormulaeEntry|page=https://formulae.org/?script=examples/Catalan_numbers}}
'''Solution'''
'''Direct definition'''
[[File:Fōrmulæ - Catalan numbers 01.png]]
[[File:Fōrmulæ - Catalan numbers 02.png]]
'''Direct definition (alternative)'''
The expression <math>\frac{(2n)!}{(n+1)!\,n!}</math> turns out to be equals to <math>\prod_{k=2}^{n}\frac{n + k}{k}</math>
[[File:Fōrmulæ - Catalan numbers 03.png]]
(same result)
'''No directly defined'''
Recursive definitions are easy to write, but extremely inefficient (specially the first one).
Because a list is intended to be get, the list of previous values can be used as a form of memoization, avoiding recursion.
The next function make use of the "second" form of recursive definition (without recursion):
[[File:Fōrmulæ - Catalan numbers 04.png]]
[[File:Fōrmulæ - Catalan numbers 05.png]]
(same result)
=={{header|GAP}}==
<
Catalan2 :=
Catalan3 := function(n)
end;
Catalan4_memo := [1];
Catalan4 := function(n)
end;
# The first fifteen: 0 to 14 !
List([0 .. 14],
List([0 .. 14],
List([0 .. 14],
List([0 .. 14],
# Same output for all four:
# [ 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440 ]</
=={{header|Go}}==
Direct:
<
import (
"fmt"
"math/big"
)
Line 1,102 ⟶ 3,031:
fmt.Println(c.Div(b.Binomial(n*2, n), c.SetInt64(n+1)))
}
}</
{{out}}
<pre>
1
Line 1,121 ⟶ 3,050:
2674440
</pre>
Recursive (alternative):
<syntaxhighlight lang="go">
package main
import (
"fmt"
"math/big"
)
func c(n int64) *big.Int {
if n == 0 {
return big.NewInt(1)
} else {
var t1, t2, t3, t4, t5, t6 big.Int
t1.Mul(big.NewInt(2), big.NewInt(n))
t2.Sub(&t1, big.NewInt(1))
t3.Mul(big.NewInt(2), &t2)
t4.Add(big.NewInt(n), big.NewInt(1))
t5.Mul(&t3, c(n-1))
t6.Div(&t5, &t4)
return &t6
}
}
func main() {
for n := int64(1); n < 16; n++ {
fmt.Println(c(n))
}
}
</syntaxhighlight>
{{out}}
<pre>
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440
9694845
</pre>
=={{header|Groovy}}==
<syntaxhighlight lang="groovy">
class Catalan
{
public static void main(String[] args)
{
BigInteger N = 15;
BigInteger k,n,num,den;
BigInteger catalan;
print(1);
for(n=2;n<=N;n++)
{
num = 1;
den = 1;
for(k=2;k<=n;k++)
{
num = num*(n+k);
den = den*k;
catalan = num/den;
}
println(catalan);
}
}
}
</syntaxhighlight>
{{out}}
<pre>
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440
9694845
</pre>
=={{header|Harbour}}==
<syntaxhighlight lang="visualfoxpro">
PROCEDURE Main()
LOCAL i
FOR i := 0 to 15
? PadL( i, 2 ) + ": " + hb_StrFormat("%d", Catalan( i ))
NEXT
RETURN
STATIC FUNCTION Catalan( n )
LOCAL i, nCatalan := 1
FOR i := 1 TO n
nCatalan := nCatalan * 2 * (2 * i - 1) / (i + 1)
NEXT
RETURN nCatalan
</syntaxhighlight>
{{out}}
<pre>
0: 1
1: 1
2: 2
3: 5
4: 14
5: 42
6: 132
7: 429
8: 1430
9: 4862
0: 16796
1: 58786
2: 208012
3: 742900
4: 2674440
5: 9694845
</pre>
=={{header|Haskell}}==
<syntaxhighlight lang="haskell">-- Three infinite lists, corresponding to the three
-- definitions in the problem statement.
cats1 :: [Integer]
cats1 =
(div . product . (enumFromTo . (2 +) <*> (2 *)))
<*> (product . enumFromTo 1) <$> [0 ..]
cats2 :: [Integer]
cats2 =
1 :
fmap
(\n -> sum (zipWith (*) (reverse (take n cats2)) cats2))
[1 ..]
cats3 :: [Integer]
cats3 =
scanl
(\c n -> c * 2 * (2 * n - 1) `div` succ n)
1
[1 ..]
main :: IO ()
main = mapM_ (print . take 15) [cats1, cats2, cats3]</syntaxhighlight>
{{out}}
<pre>[1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440]
[1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440]
[1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440]</pre>
=={{header|Icon}} and {{header|Unicon}}==
<
every writes(catalan(
end
Line 1,148 ⟶ 3,218:
static M
initial M := table()
n=0 & return 1
if n > 0 then
return (n = 1) | \M[n] | ( M[n] := (2*(2*n-1)*catalan(n-1))/(n+1))
end</
{{out}}
<pre>1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440</pre>
=={{header|J}}==
<syntaxhighlight lang="j"> ((! +:) % >:) i.15x
1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440</syntaxhighlight>
=={{header|Java}}==
Replace double inexact computations with BigInteger implementation.
<syntaxhighlight lang="java">
import java.math.BigInteger;
import java.util.ArrayList;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
public class CatlanNumbers {
public static void main(String[] args) {
Catlan f1 = new Catlan1();
Catlan f2 = new Catlan2();
Catlan f3 = new Catlan3();
System.out.printf(" Formula 1 Formula 2 Formula 3%n");
for ( int n = 0 ; n <= 15 ; n++ ) {
System.out.printf("C(%2d) = %,12d %,12d %,12d%n", n, f1.catlin(n), f2.catlin(n), f3.catlin(n));
}
}
private static interface Catlan {
public BigInteger catlin(long n);
}
private static class Catlan1 implements Catlan {
@Override
public BigInteger catlin(long n) {
List<Long> numerator = new ArrayList<>();
for ( long k = n+2 ; k <= 2*n ; k++ ) {
numerator.add(k);
}
List<Long> denominator = new ArrayList<>();
for ( long k = 2 ; k <= n ; k++ ) {
denominator.add(k);
}
for ( int i = numerator.size()-1 ; i >= 0 ; i-- ) {
for ( int j = denominator.size()-1 ; j >= 0 ; j-- ) {
if ( denominator.get(j) == 1 ) {
continue;
}
if ( numerator.get(i) % denominator.get(j) == 0 ) {
long val = numerator.get(i) / denominator.get(j);
numerator.set(i, val);
denominator.remove(denominator.get(j));
if ( val == 1 ) {
break;
}
}
}
}
BigInteger catlin = BigInteger.ONE;
for ( int i = 0 ; i < numerator.size() ; i++ ) {
catlin = catlin.multiply(BigInteger.valueOf(numerator.get(i)));
}
for ( int i = 0 ; i < denominator.size() ; i++ ) {
catlin = catlin.divide(BigInteger.valueOf(denominator.get(i)));
}
return catlin;
}
}
private static class Catlan2 implements Catlan {
private static Map<Long,BigInteger> CACHE = new HashMap<>();
static {
CACHE.put(0L, BigInteger.ONE);
}
// C(0) = 1, C(n+1) = sum(i=0..n,C(i)*C(n-i))
@Override
public BigInteger catlin(long n) {
if ( CACHE.containsKey(n) ) {
return CACHE.get(n);
}
BigInteger catlin = BigInteger.ZERO;
n--;
for ( int i = 0 ; i <= n ; i++ ) {
//System.out.println("n = " + n + ", i = " + i + ", n-i = " + (n-i));
catlin = catlin.add(catlin(i).multiply(catlin(n-i)));
}
CACHE.put(n+1, catlin);
return catlin;
}
}
private static class Catlan3 implements Catlan {
private static Map<Long,BigInteger> CACHE = new HashMap<>();
static {
CACHE.put(0L, BigInteger.ONE);
}
// C(0) = 1, C(n+1) = 2*(2n-1)*C(n-1)/(n+1)
@Override
public BigInteger catlin(long n) {
if ( CACHE.containsKey(n) ) {
return CACHE.get(n);
}
BigInteger catlin = BigInteger.valueOf(2).multiply(BigInteger.valueOf(2*n-1)).multiply(catlin(n-1)).divide(BigInteger.valueOf(n+1));
CACHE.put(n, catlin);
return catlin;
}
}
}
</syntaxhighlight>
{{out}}
<pre>
Formula 1 Formula 2 Formula 3
C( 0) = 1 1 1
C( 1) = 1 1 1
C( 2) = 2 2 2
C( 3) = 5 5 5
C( 4) = 14 14 14
C( 5) = 42 42 42
C( 6) = 132 132 132
C( 7) = 429 429 429
C( 8) = 1,430 1,430 1,430
C( 9) = 4,862 4,862 4,862
C(10) = 16,796 16,796 16,796
C(11) = 58,786 58,786 58,786
C(12) = 208,012 208,012 208,012
C(13) = 742,900 742,900 742,900
C(14) = 2,674,440 2,674,440 2,674,440
C(15) = 9,694,845 9,694,845 9,694,845
</pre>
=={{header|JavaScript}}==
===Procedural===
<syntaxhighlight lang="javascript"><html><head><title>Catalan</title></head>
<body><pre id='x'></pre><script type="application/javascript">
function disp(x) {
Line 1,189 ⟶ 3,391:
disp(i + '\t' + cata1(i) + '\t' + cata2(i) + '\t' + cata3(i));
</script></body></html></
{{out}}
<pre> meth1 meth2 meth3
0 1 1 1
1 1 1 1
Line 1,205 ⟶ 3,409:
13 742900 742900 742900
14 2674440 2674440 2674440
15 9694845 9694845 9694845</
==
Defining an infinite list:
<syntaxhighlight lang="javascript">(() => {
"use strict";
// ----------------- CATALAN NUMBERS -----------------
// catalansDefinitionThree :: [Int]
const catalansDefinitionThree = () =>
// An infinite sequence of Catalan numbers.
scanlGen(
c => n => Math.floor(
(2 * c * pred(2 * n)) / succ(n)
)
)(1)(
enumFrom(1)
);
// ---------------------- TEST -----------------------
// main :: IO ()
const main = () =>
take(15)(
catalansDefinitionThree()
);
// --------------------- GENERIC ---------------------
// enumFrom :: Enum a => a -> [a]
const enumFrom = function* (n) {
// An infinite sequence of integers,
// starting with n.
let v = n;
while (true) {
yield v;
v = 1 + v;
}
};
// pred :: Int -> Int
const pred = x =>
x - 1;
// scanlGen :: (b -> a -> b) -> b -> Gen [a] -> [b]
const scanlGen = f =>
// The series of interim values arising
// from a catamorphism over an infinite list.
startValue => function* (gen) {
let
a = startValue,
x = gen.next();
yield a;
while (!x.done) {
a = f(a)(x.value);
yield a;
x = gen.next();
}
};
// succ :: Int -> Int
const succ = x =>
1 + x;
// take :: Int -> [a] -> [a]
// take :: Int -> String -> String
const take = n =>
// The first n elements of a list,
// string of characters, or stream.
xs => Array.from({
length: n
}, () => {
const x = xs.next();
return x.done ? [] : [x.value];
}).flat();
// MAIN ---
return JSON.stringify(main(), null, 2);
})();</syntaxhighlight>
{{Out}}
<pre>[
1,
1,
2,
5,
14,
42,
132,
429,
1430,
4862,
16796,
58786,
208012,
742900,
2674440
]</pre>
=={{header|jq}}==
{{ works with|jq|1.4 }}
The recursive formula for C(n) in terms of C(n-1) lends itself
directly to efficient implementations in jq so in this section,
that formula is used (a) to define a function for computing a single Catalan number; (b) to define a function for generating a sequence of Catalan numbers; and (c) to write a single expression for generating a sequence of Catalan numbers using jq's builtin "recurse/1" filter.
==== Compute a single Catalan number====
<syntaxhighlight lang="jq">def catalan:
if . == 0 then 1
elif . < 0 then error("catalan is not defined on \(.)")
else (2 * (2*. - 1) * ((. - 1) | catalan)) / (. + 1)
end;</syntaxhighlight>
'''Example 1'''
<syntaxhighlight lang="jq">(range(0; 16), 100) as $i | $i | catalan | [$i, .]</syntaxhighlight>
{{Out}}
<div style="overflow:scroll; height:150px;">
<syntaxhighlight lang="sh">$ jq -M -n -c -f Catalan_numbers.jq
[0,1]
[1,1]
[2,2]
[3,5]
[4,14]
[5,42]
[6,132]
[7,429]
[8,1430]
[9,4862]
[10,16796]
[11,58786]
[12,208012]
[13,742900]
[14,2674440]
[15,9694845]
[100,8.96519947090131e+56]
</syntaxhighlight></div>
==== Generate a sequence of Catalan numbers ====
<syntaxhighlight lang="jq">def catalan_series(max):
def _catalan: # state: [n, catalan(n)]
if .[0] > max then empty
else .,
((.[0] + 1) as $n | .[1] as $cp
| [$n, (2 * (2*$n - 1) * $cp) / ($n + 1) ] | _catalan)
end;
[0,1] | _catalan;
</syntaxhighlight>
'''Example 2''':
<syntaxhighlight lang="jq">catalan_series(15)</syntaxhighlight>
{{Out}}
As above for 0 to 15.
==== An expression to generate Catalan numbers ====
<syntaxhighlight lang="jq">
[0,1]
| recurse( if .[0] == 15 then empty
else .[1] as $c | (.[0] + 1) | [ ., (2 * (2*. - 1) * $c) / (. + 1) ]
end )</syntaxhighlight>
{{out}}
As above for 0 to 15.
=={{header|Julia}}==
{{works with|Julia|0.6}}
<syntaxhighlight lang="julia">catalannum(n::Integer) = binomial(2n, n) ÷ (n + 1)
@show catalannum.(1:15)
@show catalannum(big(100))</syntaxhighlight>
{{out}}
<pre>catalannum.(1:15) = [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845]
catalannum(big(100)) = 896519947090131496687170070074100632420837521538745909320</pre>
(In the second example, we have used arbitrary-precision integers to avoid overflow for large Catalan numbers.)
=={{header|K}}==
<
catalan'!:15
1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440</
=={{header|Kotlin}}==
{{works with|Java|1.7.0}}
{{works with|Kotlin|1.1.4}}
<syntaxhighlight lang="scala">abstract class Catalan {
abstract operator fun invoke(n: Int) : Double
protected val m = mutableMapOf(0 to 1.0)
}
object CatalanI : Catalan() {
override fun invoke(n: Int): Double {
if (n !in m)
m[n] = Math.round(fact(2 * n) / (fact(n + 1) * fact(n))).toDouble()
return m[n]!!
}
private fun fact(n: Int): Double {
if (n in facts)
return facts[n]!!
val f = n * fact(n -1)
facts[n] = f
return f
}
private val facts = mutableMapOf(0 to 1.0, 1 to 1.0, 2 to 2.0)
}
object CatalanR1 : Catalan() {
override fun invoke(n: Int): Double {
if (n in m)
return m[n]!!
var sum = 0.0
for (i in 0..n - 1)
sum += invoke(i) * invoke(n - 1 - i)
sum = Math.round(sum).toDouble()
m[n] = sum
return sum
}
}
object CatalanR2 : Catalan() {
override fun invoke(n: Int): Double {
if (n !in m)
m[n] = Math.round(2.0 * (2 * (n - 1) + 1) / (n + 1) * invoke(n - 1)).toDouble()
return m[n]!!
}
}
fun main(args: Array<String>) {
val c = arrayOf(CatalanI, CatalanR1, CatalanR2)
for(i in 0..15) {
c.forEach { print("%9d".format(it(i).toLong())) }
println()
}
}</syntaxhighlight>
{{out}}
<pre> 1 1 1
1 1 1
2 2 2
5 5 5
14 14 14
42 42 42
132 132 132
429 429 429
1430 1430 1430
4862 4862 4862
16796 16796 16796
58786 58786 58786
208012 208012 208012
742900 742900 742900
2674440 2674440 2674440
9694845 9694845 9694845</pre>
=={{header|Lambdatalk}}==
{{trans|Javascript}}
<h3>1) catalan1</h3>
<syntaxhighlight lang="scheme">
{def catalan1
{def fac {lambda {:n} {* {S.serie 1 :n}}}}
{lambda {:n}
{floor {+ {/ {fac {* 2 :n}} {fac {+ :n 1}} {fac :n}} 0.5}}}}
-> catalan1
{S.map catalan1 {S.serie 1 15}}
-> 1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845
</syntaxhighlight>
<h3>2) catalan2</h3>
<syntaxhighlight lang="scheme">
{def catalan2
{def catalan2.sum
{lambda {:n :a :s :i}
{if {= :i :n}
then {A.set! :n :s :a}
else {catalan2.sum :n
:a
{+ :s {* {catalan2.loop :i :a}
{catalan2.loop {- :n :i 1} :a}}}
{+ :i 1}} }}}
{def catalan2.loop
{lambda {:n :a}
{if {= :n 0}
then 1
else {if {W.equal? {A.get :n :a} undefined}
then {A.get :n {catalan2.sum :n :a 0 0}}
else {A.get :n :a} }}}}
{lambda {:n}
{catalan2.loop :n {A.new}} }}
-> catalan2
{S.map catalan2 {S.serie 0 15}}
-> 1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845
</syntaxhighlight>
<h3>3) catalan3</h3>
<syntaxhighlight lang="scheme">
{def catalan3
{def catalan3.loop
{lambda {:n :a}
{if {= :n 0}
then 1
else {if {W.equal? {A.get :n :a} undefined}
then {A.get :n
{A.set! :n
{/ {* {- {* 4 :n} 2}
{catalan3.loop {- :n 1} :a}}
{+ :n 1}}
:a}}
else {A.get :n :a}
}}}}
{lambda {:n}
{catalan3.loop :n {A.new}}}}
-> catalan3
{S.map catalan3 {S.serie 0 15}}
-> 1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845
</syntaxhighlight>
<h3>4) Alternative for a vertical diplay</h3>
<syntaxhighlight lang="scheme">
{style
td { text-align:right;
font-family:monospace;
}
}
{table
{tr {td} {td cat1} {td cat2} {td cat3}}
{S.map {lambda {:i} {tr {td :i}
{td {catalan1 :i}}
{td {catalan2 :i}}
{td {catalan3 :i}}}}
{S.serie 0 15}}
}
cat1 cat2 cat3
0 1 1 1
1 1 1 1
2 2 2 2
3 5 5 5
4 14 14 14
5 42 42 42
6 132 132 132
7 429 429 429
8 1430 1430 1430
9 4862 4862 4862
10 16796 16796 16796
11 58786 58786 58786
12 208012 208012 208012
13 742900 742900 742900
14 2674440 2674440 2674440
15 9694845 9694845 9694845
</syntaxhighlight>
=={{header|langur}}==
{{trans|Perl}}
<syntaxhighlight lang="langur">val factorial = fn x:if(x < 2: 1; x * self(x - 1))
val catalan = fn n:factorial(2 * n) / factorial(n+1) / factorial(n)
for i in 0..15 {
writeln "{{i:2}}: {{catalan(i):10}}"
}
writeln "10000: ", catalan(10000)
</syntaxhighlight>
{{out}}
<pre> 0: 1
1: 1
2: 2
3: 5
4: 14
5: 42
6: 132
7: 429
8: 1430
9: 4862
10: 16796
11: 58786
12: 208012
13: 742900
14: 2674440
15: 9694845
10000: 224537812493385215633593584257360578701103586219365887773293713835854436588700534490998102719114320210209905393799589701149327326500953702713977513001838761306936534407802585494454599941773729984591764542782202886796997833276495496514760245912220654267091568311812071300891219894022165175451441066691435091975969499731921675488934120638046514134965974069039677192984714638704528752769863567952620334847707274529741976558104236293861846622622783294667505268651205024766408784881872997404042356319626323351089169906635603513309014645157443570842822082866699012415455339518777770781742052837799476906230350785959040487158118992753484022865373274100095762968510625236915280143408460651206678398725681703811505423791566261735329550627967717189932855983913468867794806585863794483869239933179341394259456515091026456652770409848702116046445406995085092488210998732255656992243441519938747425554228724734242623566663631968254490897214106655375215196762710825001305055093871863518797311135688370964194817463890187212845332422257193414201244344808864449873736345425670715824582633802476282521798739438044652622163657359012681653473214512797365047989922327391063907061792126264420963262176161781711086630089636828211837643128677915076724947168653050318426339007489738275045346257959685376480042860870398232333705506506342394485443047987642390287346746539674780326188825579548593281319807827279403944008553690033855132088140116099772393778770685018936338194366302053586633406848404622048675525765095697363909787189635178694239275237185046710057476484117945279786897787624602379494797322427251542758312638233073625857897083435831841717971137851874666094337671443717108457737153283641719103639784923520519013700030680553564442331411313831920775983175313709250333784211385811480015293165463406576311626295629412110652218717603537723650144357966952842696678735624157616428716812764985074925414219421312810089785108621126934245959900367104035334200067714905754827856122801987429837706493130435832752072139392743006620396370486473952500144779413596417260472218266529167783118015414918168260722824885550181735638670588682513610805160133611349864194033776132438535863120087679096358696928233598996870302136347936567444208209125300149683552369341937471817860835774359234009557030148123353114950735217736514617017504851011193104728986836180908987352236659629183725016607437110422583156042941955830763092095074443334625318588569114114087985404048889671202396824806275701581378689568449507132793603852731445602923990458926101180821029108808623323378547869169352237448925371763574346501610378415722137519019474474794069155118626291447578558908522430436148987521551911541787974276591708584289036595642180860178815462862735993859177180582760389253540408842580225467216988321950591728369194164290645992782274919561096308372635908842325870580231011459216934235078490764707633348336131667313582584404397290232519769625777374165187949140092779343812345117947306771376053099536367169631889642304360871187460737580808157222861127968703067542270175460553478533349238111434409526724363429611803844595968793121871649699680963646793415774160274520010905236593324062464542927011227158945796188186430711399250096518886617184049325827319276468018789191520522185358895653192882843061349706085770767046601045697944646638311930027354235643643713545212361580694059553720806659066661496416423676930095857438882302891350789287291844752601744462789158506243012088536936184422120232369244564444689340142897415432231452353338115944183447986470689449043710051589958391273681116292415738776171575775695905846247205522469202801517417551374761549677412720803623129527503286287755308576386461385928958587649159872019202866614901547860974883963007792442796064165417207167072370586790722366932349325253877744621251386864069101337572557790214048760202008337611577675840153696735860276810033694744314488435390547908483357054897387317002405793108554524629034558098886977538473481750772616164313845337139245688079995996839933620829828339492800825536599964878893947278408890351634126931068657027524005795713514365098086505030570362785115155293306343520969872400876180105031975302255898787642403303027682634969586730202117121076117629457710028105378124677420093990476071697970354661002217702623344454780740808459286778553016318604430682610618871098652904537323336381304469735192868285840882036271136058499391069436145426450229039329475974178236465920534171895204155964515055983303017823692138977622016292722019365841360360274557488926673754175222061483328914099598663902320310143583379354121664996173733086613692927391384486261610892314450463841637667054196985332620403539011932606618414419229492637564924726411270720189611019154677281846409387514072618176832310721327819277699943226895919915049652045449281057471199978267843961724883768772155477073354744908923995448752333726740642292872107500458349718026322755698226793850983280706045951407323891263270928264657562125955511946782954645656015480418543664557515041692091317941000997342935512311493290722434384401250133402934163457264794261787386862382738330195237770190998115114193014769006071380834085352290585937952429981509893303796306071520571655936820282768086579891336876000368502562579738337809071051261343359121744773055264455701014137255399929760233753812017596045145926791136761130783810840502248142803073720015451941006030172192834375431286154255159659778817089767964922549014569972777126726537787896968876337799235679125368824867754881036161730805613471278633981478858113141202728303435218970292775366288829203013873713349923690394124920402725698544786016048685431525811047414746045227535216327530901827040588505255466803793791888002231571686068617764292584075135236237044383334893874602177596602979234717936820827427229615827657960492946059695301906791494260652411424538532836730097985187522379068364429583532675896349363295120431429006688249818006722311568902288350452581968418068616818268667067741994472455501649753611708445979082338902214467454627107888156489438584617793175431865532382711812960546611287516640
</pre>
=={{header|Logo}}==
<syntaxhighlight lang="logo">to factorial :n
output ifelse [less? :n 1] 1 [product :n factorial difference :n 1]
end
to choose :n :r
output quotient factorial :n product factorial :r factorial difference :n :r
end
to catalan :n
output product (quotient sum :n 1) choose product 2 :n :n
end
repeat 15 [print catalan repcount]</syntaxhighlight>
{{out}}
<pre>
1
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440</pre>
=={{header|Logtalk}}==
For this task it is instructive to show a more general-purpose interface for sequences and an implementation of it for Catalan numbers.
First, <code>loader.lgt</code>:
<syntaxhighlight lang="logtalk">
:- initialization((
% libraries
logtalk_load(dates(loader)),
logtalk_load(meta(loader)),
logtalk_load(types(loader)),
% application
logtalk_load(seqp),
logtalk_load(catalan),
logtalk_load(catalan_test)
)).
</syntaxhighlight>
The interface is defined in <code>seqp.lgt</code> as a protocol:
<syntaxhighlight lang="logtalk">
:- protocol(seqp).
:- public(init/0). % reset to a beginning state if meaningful
:- public(nth/2). % get the nth value of the sequence
:- public(to_nth/2). % get from the start to the nth value of the sequence as a list
:- end_protocol.
</syntaxhighlight>
The implementation of a Catalan sequence generator is in <code>catalan.lgt</code>:
<syntaxhighlight lang="logtalk">
:- object(catalan, implements(seqp)).
:- private(catalan/2).
:- dynamic(catalan/2).
% Public interface.
init :- retractall(catalan(_,_)). % flush any memoized results
nth(N, V) :- \+ catalan(N, V), catalan_(N, V), !. % generate iff it's not been memoized
nth(N, V) :- catalan(N, V), !. % otherwise use the memoized version
to_nth(N, L) :-
integer::sequence(0, N, S), % generate a list of 0 to N
meta::map(nth, S, L). % map the nth/2 predicate to the list for all Catalan numbers up to N
% Local helper predicates.
catalan_(N, V) :-
N > 0, % calculate
N1 is N - 1,
N2 is N + 1,
catalan_(N1, V1), % via a recursive call
V is V1 * 2 * (2 * N - 1) // N2,
assertz(catalan(N, V)). % and memoize the result
catalan_(0, 1).
:- end_object.
</syntaxhighlight>
This is a memoizing implementation whose impact we will check in the test. The <code>init/0</code> predicate flushes any memoized results.
The test driver is a simple one that generates the first fifteen Catalan numbers four times, comparing times with and without memoization. From <code>catalan_test.lgt</code>:
<syntaxhighlight lang="logtalk">
:- object(catalan_test).
:- public(run/0).
run :-
% put the object into a known initial state
catalan::init,
% first 15 Catalan numbers, record duration.
time_operation(catalan::to_nth(15, C1), D1),
% first 15 Catalan numbers again, twice, recording duration.
time_operation(catalan::to_nth(15, C2), D2),
time_operation(catalan::to_nth(15, C3), D3),
% reset the object again
catalan::init,
% first 15 Catalan numbers, record duration.
time_operation(catalan::to_nth(15, C4), D4),
% ensure the results were the same each time
C1 = C2, C2 = C3, C3 = C4,
% write the results and durations of each run
write(C1), write(' '), write(D1), nl,
write(C2), write(' '), write(D2), nl,
write(C3), write(' '), write(D3), nl,
write(C4), write(' '), write(D4), nl.
% visual inspection should show all results the same
% first and final durations should be much larger
:- meta_predicate(time_operation(0, *)).
time_operation(Goal, Duration) :-
time::cpu_time(Before),
call(Goal),
time::cpu_time(After),
Duration is After - Before.
:- end_object.
</syntaxhighlight>
{{Out}}
The session at the top-level looks like this:
<pre>
?- {loader}.
% ... messages elided ...
% (0 warnings)
true.
?- catalan_test::run.
[1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440,9694845] 0.001603
[1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440,9694845] 0.000306
[1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440,9694845] 0.00026
[1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440,9694845] 0.001346
true.
</pre>
This test shows:
# The <code>nth/2</code> predicate works (since <code>to_nth/2</code> is implemented in terms of it).
# The <code>to_nth/2</code> predicate works.
# Memoization generates a speedup of between ~4.5× to ~6.2× over generating from scratch.
=={{header|Lua}}==
<syntaxhighlight lang="lua">-- recursive with memoization
local catalan = { [0] = 1 }
setmetatable(catalan, {
__index = function(c, n)
c[n] = c[n - 1] * 2 * (2 * n - 1) / (n + 1)
return c[n]
end
})
for i = 0, 14 do
print(string.format("%d", catalan[i]))
end</syntaxhighlight>
{{out}}
<pre>1
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440</pre>
=={{header|MAD}}==
<syntaxhighlight lang="mad"> NORMAL MODE IS INTEGER
DIMENSION C(15)
C(0) = 1
THROUGH CALC, FOR N=1, 1, N.GE.15
CALC C(N) = ((4*N-2)*C(N-1))/(N+1)
THROUGH SHOW, FOR N=0, 1, N.GE.15
SHOW PRINT FORMAT CFMT,N,C(N)
VECTOR VALUES CFMT=$2HC(,I2,4H) = ,I7*$
END OF PROGRAM </syntaxhighlight>
{{out}}
<pre>C( 0) = 1
C( 1) = 1
C( 2) = 2
C( 3) = 5
C( 4) = 14
C( 5) = 42
C( 6) = 132
C( 7) = 429
C( 8) = 1430
C( 9) = 4862
C(10) = 16796
C(11) = 58786
C(12) = 208012
C(13) = 742900
C(14) = 2674440</pre>
=={{header|Maple}}==
<syntaxhighlight lang="maple">CatalanNumbers := proc( n::posint )
return seq( (2*i)!/((i + 1)!*i!), i = 0 .. n - 1 );
end proc:
CatalanNumbers(15);
</syntaxhighlight>
Output:
<pre>
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440
</pre>
=={{header|Mathematica}} / {{header|Wolfram Language}}==
<syntaxhighlight lang="mathematica">CatalanN[n_Integer /; n >= 0] := (2 n)!/((n + 1)! n!)</syntaxhighlight>
{{out|Sample Output}}
<syntaxhighlight lang="mathematica">TableForm[CatalanN/@Range[0,15]]
//TableForm=
1
Line 1,351 ⟶ 4,068:
742900
2674440
9694845</syntaxhighlight>
=={{header|MATLAB}} / {{header|Octave}}==
<syntaxhighlight lang="matlab">function n = catalanNumber(n)
for i = (1:length(n))
n(i) = (1/(n(i)+1))*nchoosek(2*n(i),n(i));
end
end</
The following version computes at the same time the n first Catalan numbers (including C0).
<
n = [1 cumprod((2:4:4*n-6) ./ (2:n))];
end</syntaxhighlight>
{{out|Sample Output}}
<syntaxhighlight lang="matlab">>> catalanNumber(14)
ans =
Line 1,367 ⟶ 4,086:
2674440
>> catalanNumbers(
ans =
Line 1,388 ⟶ 4,107:
9694845
35357670
129644790</
The following version uses the identity Ln(x!)=Gammaln(x+1) and prod(1:x)=sum(ln(1:x))
<syntaxhighlight lang="matlab">
CatalanNumber=@(n) round(exp(gammaln(2*n+1)-sum(gammaln([n+2 n+1]))));
</syntaxhighlight>
{{out|Sample Output}}
<syntaxhighlight lang="matlab">>>CatalanNumber(10)
ans =
16796
>> num2str(CatalanNumber(20))
ans =
'6564120420'
</syntaxhighlight>
=={{header|Maxima}}==
<syntaxhighlight lang="maxima">/* The following is an array function, hence the square brackets. It uses memoization automatically */
cata[n] := sum(cata[i]*cata[n - 1 - i], i, 0, n - 1)$
cata[0]: 1$
cata2(n) := binomial(2*n, n)/(n + 1)$
makelist(cata[n], n, 0, 14);
makelist(cata2(n), n, 0, 14);
/* both return [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440] */</syntaxhighlight>
=={{header|Miranda}}==
<syntaxhighlight lang="miranda">main :: [sys_message]
main = [Stdout (lay (map (show . catalan) [0..14]))]
catalan :: num->num
catalan 0 = 1
catalan n = (4*n - 2) * catalan (n - 1) div (n + 1)</syntaxhighlight>
{{out}}
<pre>1
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440</pre>
=={{header|Modula-2}}==
<syntaxhighlight lang="modula2">MODULE CatalanNumbers;
FROM FormatString IMPORT FormatString;
FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
PROCEDURE binomial(m,n : LONGCARD) : LONGCARD;
VAR r,d : LONGCARD;
BEGIN
r := 1;
d := m - n;
IF d>n THEN
n := d;
d := m - n;
END;
WHILE m>n DO
r := r * m;
DEC(m);
WHILE (d>1) AND NOT (r MOD d # 0) DO
r := r DIV d;
DEC(d)
END
END;
RETURN r
END binomial;
PROCEDURE catalan1(n : LONGCARD) : LONGCARD;
BEGIN
RETURN binomial(2*n,n) DIV (1+n)
END catalan1;
PROCEDURE catalan2(n : LONGCARD) : LONGCARD;
VAR i,sum : LONGCARD;
BEGIN
IF n>1 THEN
sum := 0;
FOR i:=0 TO n-1 DO
sum := sum + catalan2(i) * catalan2(n - 1 - i)
END;
RETURN sum
ELSE
RETURN 1
END
END catalan2;
PROCEDURE catalan3(n : LONGCARD) : LONGCARD;
BEGIN
IF n#0 THEN
RETURN 2 *(2 * n - 1) * catalan3(n - 1) DIV (1 + n)
ELSE
RETURN 1
END
END catalan3;
VAR
blah : LONGCARD = 123;
buf : ARRAY[0..63] OF CHAR;
i : LONGCARD;
BEGIN
FormatString("\tdirect\tsumming\tfrac\n", buf);
WriteString(buf);
FOR i:=0 TO 15 DO
FormatString("%u\t%u\t%u\t%u\n", buf, i, catalan1(i), catalan2(i), catalan3(i));
WriteString(buf)
END;
ReadChar
END CatalanNumbers.</syntaxhighlight>
=={{header|Nim}}==
<syntaxhighlight lang="nim">import math
import strformat
proc catalan1(n: int): int =
binom(2 * n, n) div (n + 1)
proc catalan2(n: int): int =
if n == 0:
return 1
for i in 0..<n:
result += catalan2(i) * catalan2(n - 1 - i)
proc catalan3(n: int): int =
if n > 0: 2 * (2 * n - 1) * catalan3(n - 1) div (1 + n)
else: 1
for i in 0..15:
echo &"{i:7} {catalan1(i):7} {catalan2(i):7} {catalan3(i):7}"</syntaxhighlight>
{{out}}
<pre> 0 1 1 1
1 1 1 1
2 2 2 2
3 5 5 5
4 14 14 14
5 42 42 42
6 132 132 132
7 429 429 429
8 1430 1430 1430
9 4862 4862 4862
10 16796 16796 16796
11 58786 58786 58786
12 208012 208012 208012
13 742900 742900 742900
14 2674440 2674440 2674440
15 9694845 9694845 9694845</pre>
=={{header|OCaml}}==
<syntaxhighlight lang="ocaml">let imp_catalan n =
let return = ref 1 in
for i = 1 to n do
return := !return * 2 * (2 * i - 1) / (i + 1)
done;
!return
let rec catalan = function
| 0 -> 1
| n -> catalan (n - 1) * 2 * (2 * n - 1) / (n + 1)
let memoize f =
let cache = Hashtbl.create 20 in
fun n ->
match Hashtbl.find_opt cache n with
| None ->
let x = f n in
Hashtbl.replace cache n x;
x
| Some x -> x
let catalan_cache = Hashtbl.create 20
let rec memo_catalan n =
if n = 0 then 1 else
match Hashtbl.find_opt catalan_cache n with
| None ->
let x = memo_catalan (n - 1) * 2 * (2 * n - 1) / (n + 1) in
Hashtbl.replace catalan_cache n x;
x
| Some x -> x
let () =
if not !Sys.interactive then
let bench label f n times =
let start = Unix.gettimeofday () in
begin
for i = 1 to times do f n done;
let stop = Unix.gettimeofday () in
Printf.printf "%s (%d runs) : %.3f\n"
label times (stop -. start)
end in
let show f g h f' n =
for i = 0 to n do
Printf.printf "%2d %7d %7d %7d %7d\n"
i (f i) (g i) (h i) (f' i)
done
in
List.iter (fun (l, f) -> bench l f 15 10_000_000)
["imperative", imp_catalan;
"recursive", catalan;
"hand-memoized", memo_catalan;
"memoized", (memoize catalan)];
show imp_catalan catalan memo_catalan (memoize catalan) 15
</syntaxhighlight>
{{out}}
<pre>$ ocaml unix.cma catalan.ml
imperative (10000000 runs) : 3.420
recursive (10000000 runs) : 3.821
hand-memoized (10000000 runs) : 0.531
memoized (10000000 runs) : 0.515
0 1 1 1 1
1 1 1 1 1
2 2 2 2 2
3 5 5 5 5
4 14 14 14 14
5 42 42 42 42
6 132 132 132 132
7 429 429 429 429
8 1430 1430 1430 1430
9 4862 4862 4862 4862
10 16796 16796 16796 16796
11 58786 58786 58786 58786
12 208012 208012 208012 208012
13 742900 742900 742900 742900
14 2674440 2674440 2674440 2674440
15 9694845 9694845 9694845 9694845
$ ocamlopt -O2 unix.cmxa catalan.ml -o catalan
$ ./catalan
imperative (10000000 runs) : 2.020
recursive (10000000 runs) : 2.283
hand-memoized (10000000 runs) : 0.159
memoized (10000000 runs) : 0.167
...</pre>
=={{header|Oforth}}==
<syntaxhighlight lang="oforth">: catalan( n -- m )
n ifZero: [ 1 ] else: [ catalan( n 1- ) 2 n * 1- * 2 * n 1+ / ] ;</syntaxhighlight>
{{out}}
<pre>
import: mapping
seqFrom(0, 15) map( #catalan ) .
[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845]
</pre>
=={{header|ooRexx}}==
Three versions of this.
<syntaxhighlight lang="oorexx">loop i = 0 to 15
say "catI("i") =" .catalan~catI(i)
say "catR1("i") =" .catalan~catR1(i)
say "catR2("i") =" .catalan~catR2(i)
end
-- This is implemented as static members on a class object
-- so that the code is able to keep state information between calls. This
-- memoization will speed up things like factorial calls by remembering previous
-- results.
::class catalan
-- initialize the class object
::method init class
expose facts catI catR1 catR2
facts = .table~new
catI = .table~new
catR1 = .table~new
catR2 = .table~new
-- seed a few items
facts[0] = 1
facts[1] = 1
facts[2] = 2
catI[0] = 1
catR1[0] = 1
catR2[0] = 1
-- private factorial method
::method fact private class
expose facts
use arg n
-- see if we've calculated this before
if facts~hasIndex(n) then return facts[n]
numeric digits 120
fact = 1
loop i = 2 to n
fact *= i
end
-- save this result
facts[n] = fact
return fact
::method catI class
expose catI
use arg n
numeric digits 20
res = catI[n]
if res == .nil then do
-- dividing by 1 removes insignificant trailing 0s
res = (self~fact(2 * n)/(self~fact(n + 1) * self~fact(n))) / 1
catI[n] = res
end
return res
::method catR1 class
expose catR1
use arg n
numeric digits 20
if catR1~hasIndex(n) then return catR1[n]
sum = 0
loop i = 0 to n - 1
sum += self~catR1(i) * self~catR1(n - 1 - i)
end
-- remove insignificant trailing 0s
sum = sum / 1
catR1[n] = sum
return sum
::method catR2 class
expose catR2
use arg n
numeric digits 20
res = catR2[n]
if res == .nil then do
res = ((2 * (2 * n - 1) * self~catR2(n - 1)) / (n + 1))
catR2[n] = res
end
return res</syntaxhighlight>
{{out}}
<pre>catI(0) = 1
catR1(0) = 1
catR2(0) = 1
catI(1) = 1
catR1(1) = 1
catR2(1) = 1
catI(2) = 2
catR1(2) = 2
catR2(2) = 2
catI(3) = 5
catR1(3) = 5
catR2(3) = 5
catI(4) = 14
catR1(4) = 14
catR2(4) = 14
catI(5) = 42
catR1(5) = 42
catR2(5) = 42
catI(6) = 132
catR1(6) = 132
catR2(6) = 132
catI(7) = 429
catR1(7) = 429
catR2(7) = 429
catI(8) = 1430
catR1(8) = 1430
catR2(8) = 1430
catI(9) = 4862
catR1(9) = 4862
catR2(9) = 4862
catI(10) = 16796
catR1(10) = 16796
catR2(10) = 16796
catI(11) = 58786
catR1(11) = 58786
catR2(11) = 58786
catI(12) = 208012
catR1(12) = 208012
catR2(12) = 208012
catI(13) = 742900
catR1(13) = 742900
catR2(13) = 742900
catI(14) = 2674440
catR1(14) = 2674440
catR2(14) = 2674440
catI(15) = 9694845
catR1(15) = 9694845
catR2(15) = 9694845</pre>
=={{header|PARI/GP}}==
Memoization is not worthwhile; PARI has fast built-in facilities for calculating binomial coefficients and factorials.
<syntaxhighlight lang="parigp">catalan(n)=binomial(2*n,n+1)/n</syntaxhighlight>
A second version:
<syntaxhighlight lang="parigp">catalan(n)=(2*n)!/(n+1)!/n!</syntaxhighlight>
Naive version with binary splitting:
<syntaxhighlight lang="parigp">catalan(n)=prod(k=n+2,2*n,k)/prod(k=2,n,k)</syntaxhighlight>
Naive version:
<syntaxhighlight lang="parigp">catalan(n)={
my(t=1);
for(k=n+2,2*n,t*=k);
for(k=2,n,t/=k);
t
};</syntaxhighlight>
The first version takes about 1.5 seconds to compute the millionth Catalan number, while the second takes 3.9 seconds. The naive implementations, for comparison, take 21 and 45 minutes. In any case, printing the first 15 is simple:
<syntaxhighlight lang="parigp">vector(15,n,catalan(n))</syntaxhighlight>
=={{header|Pascal}}==
<
function catalanNumber1(n: integer): double;
Line 1,422 ⟶ 4,530:
for number := 0 to 14 do
writeln (number:3, round(catalanNumber1(number)):9);
end.</
{{out}}
<pre>
:> ./CatalanNumbers
Line 1,444 ⟶ 4,552:
14 2674440
</pre>
=={{header|Perl}}==
<
sub catalan {
my $n = shift;
factorial(2*$n) / factorial($n+1) / factorial($n);
}
print "$_\t@{[ catalan($_) ]}\n" for 0 .. 20;</syntaxhighlight>
For computing up to 20 ish, memoization is not needed. For much bigger numbers, this is faster:
<
sub catalan {
use bigint;
Line 1,459 ⟶ 4,568:
# most of the time is spent displaying the long numbers, actually
print "$_\t", catalan($_), "\n" for 0 .. 10000;</
That has two downsides: high memory use and slow access to an isolated large value. Using a fast binomial function can solve both these issues. The downside here is if the platform doesn't have the GMP library then binomials won't be fast.
{{libheader|ntheory}}
<syntaxhighlight lang="perl">use ntheory qw/binomial/;
sub catalan {
my $n = shift;
binomial(2*$n,$n)/($n+1);
}
print "$_\t", catalan($_), "\n" for 0 .. 10000;</syntaxhighlight>
=={{header|Phix}}==
See also [[Catalan_numbers/Pascal%27s_triangle#Phix]] which may be faster.
<!--<syntaxhighlight lang="phix">(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #000080;font-style:italic;">-- returns inf/-nan for n>85, and needs the rounding for n>=14, accurate to n=29</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">catalan1</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">factorial</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)/(</span><span style="color: #7060A8;">factorial</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)*</span><span style="color: #7060A8;">factorial</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">))+</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #000080;font-style:italic;">-- returns inf for n>519, accurate to n=30:</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">catalan2</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- NB: very slow!</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">not</span> <span style="color: #000000;">n</span>
<span style="color: #000000;">n</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">res</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">catalan2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)*</span><span style="color: #000000;">catalan2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">res</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #000080;font-style:italic;">-- returns inf for n>514, accurate to n=30:</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">catalan3</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">1</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)/(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">+</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)*</span><span style="color: #000000;">catalan3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">4</span><span style="color: #0000FF;">),</span><span style="color: #000000;">16</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">times</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">4</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">t0</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #000000;">15</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">switch</span> <span style="color: #000000;">t</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">case</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">:</span> <span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">][</span><span style="color: #000000;">2</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">catalan1</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">case</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">:</span> <span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">][</span><span style="color: #000000;">3</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">catalan2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">case</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">:</span> <span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">][</span><span style="color: #000000;">4</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">catalan3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">case</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">:</span> <span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">][</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">;</span> <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%2d: %10d %10d %10d\n"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">res</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">switch</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">=</span><span style="color: #000000;">4</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">times</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">elapsed</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">time</span><span style="color: #0000FF;">()-</span><span style="color: #000000;">t0</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"times:%8s %10s %10s\n"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">times</span><span style="color: #0000FF;">)</span>
<!--</syntaxhighlight>-->
{{out}}
<pre>
0: 1 1 1
1: 1 1 1
2: 2 2 2
3: 5 5 5
4: 14 14 14
5: 42 42 42
6: 132 132 132
7: 429 429 429
8: 1430 1430 1430
9: 4862 4862 4862
10: 16796 16796 16796
11: 58786 58786 58786
12: 208012 208012 208012
13: 742900 742900 742900
14: 2674440 2674440 2674440
15: 9694845 9694845 9694845
times: 0s 1.6s 0s
</pre>
As expected, catalan2() is by far the slowest, so let's memoise that one!
=== memoized recursive gmp version ===
{{libheader|Phix/mpfr}}
<!--<syntaxhighlight lang="phix">(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">include</span> <span style="color: #000000;">builtins</span><span style="color: #0000FF;">\</span><span style="color: #004080;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">c2cache</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">catalan2m</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- very fast!</span>
<span style="color: #004080;">object</span> <span style="color: #000000;">r</span> <span style="color: #000080;font-style:italic;">-- result (a [cached/shared] mpz)
-- (nb: modifying result will mess up cache)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;"><=</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c2cache</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">c2cache</span><span style="color: #0000FF;">[</span><span style="color: #000000;">n</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">r</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">else</span>
<span style="color: #000000;">c2cache</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c2cache</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">mpz</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_init</span><span style="color: #0000FF;">()</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #7060A8;">mpz_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">catalan2m</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">),</span><span style="color: #000000;">catalan2m</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">-</span><span style="color: #000000;">i</span><span style="color: #0000FF;">))</span>
<span style="color: #7060A8;">mpz_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_free</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">c2cache</span><span style="color: #0000FF;">[</span><span style="color: #000000;">n</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">r</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">r</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #000000;">15</span> <span style="color: #008080;">do</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">catalan2m</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)))</span> <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"0..15: %s\n"</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">join</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span><span style="color: #008000;">","</span><span style="color: #0000FF;">))</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"100: %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">catalan2m</span><span style="color: #0000FF;">(</span><span style="color: #000000;">100</span><span style="color: #0000FF;">))})</span>
<!--</syntaxhighlight>-->
{{out}}
<pre>
0..15: 1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440,9694845
100: 896519947090131496687170070074100632420837521538745909320
</pre>
=={{header|PHP}}==
<syntaxhighlight lang="php"><?php
class CatalanNumbersSerie
Line 1,519 ⟶ 4,714:
echo "$i = $r\r\n";
}
?></syntaxhighlight>
{{out}}
<pre>
0 = 1
Line 1,541 ⟶ 4,734:
15 = 9694845
</pre>
<syntaxhighlight lang="php">
<?php
$n = 15;
$t[1] = 1;
foreach (range(1, $n+1) as $i) {
foreach (range($i, 1-1) as $j) {
$t[$j] += $t[$j - 1];
}
$t[$i +1] = $t[$i];
foreach (range($i+1, 1-1) as $j) {
$t[$j] += $t[$j -1];
}
print ($t[$i+1]-$t[$i])."\t";
}
</syntaxhighlight>
{{out}}
<pre>
1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845 35357670
</pre>
=={{header|Picat}}==
{{works with|Picat}}
<syntaxhighlight lang="picat">
table
factorial(0) = 1.
factorial(N) = N * factorial(N - 1).
catalan1(N) = factorial(2 * N) // (factorial(N + 1) * factorial(N)).
catalan2(0) = 1.
catalan2(N) = 2 * (2 * N - 1) * catalan2(N - 1) // (N + 1).
main =>
foreach (I in 0..14)
printf("%d. %d = %d\n", I, catalan1(I), catalan2(I))
end.
</syntaxhighlight>
{{out}}
<pre>
0. 1 = 1
1. 1 = 1
2. 2 = 2
3. 5 = 5
4. 14 = 14
5. 42 = 42
6. 132 = 132
7. 429 = 429
8. 1430 = 1430
9. 4862 = 4862
10. 16796 = 16796
11. 58786 = 58786
12. 208012 = 208012
13. 742900 = 742900
14. 2674440 = 2674440
</pre>
=={{header|PicoLisp}}==
<
(de fact (N)
(if (=0 N)
Line 1,557 ⟶ 4,805:
(if (=0 N)
1
(cache '(NIL)
(sum
'((I) (* (catalanRec I) (catalanRec (- N I 1))))
Line 1,575 ⟶ 4,823:
(catalanDir N)
(catalanRec N)
(catalanAlt N) ) )</
{{out}}
<pre> 0 => 1 1 1
1 => 1 1 1
Line 1,593 ⟶ 4,841:
14 => 2674440 2674440 2674440</pre>
=={{header|PL/0}}==
{{trans|Tiny BASIC}}
Integers are limited to 32767 so only the first ten Catalan numbers can be represented. To avoid internal overflow, the program subtracts something clever from <code>c</code> and then adds it back at the end.
<syntaxhighlight lang="pascal">
var n, c, i;
begin
n := 0; c := 1;
! c;
while n <= 9 do
begin
n := n + 1;
i := 0;
while c > 0 do
begin
c := c - (n + 1);
i := i + 1
end;
c := 2 * (2 * n - 1) * c / (n + 1);
c := c + 2 * i * (2 * n - 1);
! c
end;
end.
</syntaxhighlight>
{{out}}
<pre>
1
1
2
5
14
42
132
429
1430
4862
16796
</pre>
=={{header|PL/I}}==
<syntaxhighlight lang="pl/i">catalan: procedure options (main); /* 23 February 2012 */
declare (i, n) fixed;
put skip list ('How many catalan numbers do you want?');
get list (n);
do i = 0 to n;
put skip list (c(i));
end;
c: procedure (n) recursive returns (fixed decimal (15));
declare n fixed;
if n <= 1 then return (1);
return ( 2*(2*n-1) * c(n-1) / (n + 1) );
end c;
end catalan;</syntaxhighlight>
{{out}}
<pre>
How many catalan numbers do you want?
1
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440
9694845
35357670
129644790
477638700
1767263190
6564120420
</pre>
=={{header|PlainTeX}}==
<syntaxhighlight lang="tex">\newcount\n
\newcount\r
\newcount\x
\newcount\ii
\def\catalan#1{%
\n#1\advance\n by1\ii1\r1%
\loop{%
\x\ii%
\multiply\x by 2 \advance\x by -1 \multiply\x by 2%
\global\multiply\r by\x%
\global\advance\ii by1%
\global\divide\r by\ii%
} \ifnum\number\ii<\n\repeat%
\the\r
}
\rightskip=0pt plus1fil\parindent=0pt
\loop{${\rm Catalan}(\the\x) = \catalan{\the\x}$\hfil\break}%
\advance\x by 1\ifnum\x<15\repeat
\bye</syntaxhighlight>
=={{header|PowerShell}}==
<syntaxhighlight lang="powershell">
function Catalan([uint64]$m) {
function fact([bigint]$n) {
if($n -lt 2) {[bigint]::one}
else{2..$n | foreach -Begin {$prod = [bigint]::one} -Process {$prod = [bigint]::Multiply($prod,$_)} -End {$prod}}
}
$fact = fact $m
$fact1 = [bigint]::Multiply($m+1,$fact)
[bigint]::divide((fact (2*$m)), [bigint]::Multiply($fact,$fact1))
}
0..15 | foreach {"catalan($_): $(catalan $_)"}
</syntaxhighlight>
<b>Output:</b>
<pre>
catalan(0): 1
catalan(1): 1
catalan(2): 2
catalan(3): 5
catalan(4): 14
catalan(5): 42
catalan(6): 132
catalan(7): 429
catalan(8): 1430
catalan(9): 4862
catalan(10): 16796
catalan(11): 58786
catalan(12): 208012
catalan(13): 742900
catalan(14): 2674440
catalan(15): 9694845
</pre>
===An Alternate Version===
This version could easily be modified to work with big integers.
<syntaxhighlight lang="powershell">
function Get-CatalanNumber
{
[CmdletBinding()]
[OutputType([PSCustomObject])]
Param
(
[Parameter(Mandatory=$true,
ValueFromPipeline=$true,
ValueFromPipelineByPropertyName=$true,
Position=0)]
[uint32[]]
$InputObject
)
Begin
{
function Get-Factorial ([int]$Number)
{
if ($Number -eq 0)
{
return 1
}
$factorial = 1
1..$Number | ForEach-Object {$factorial *= $_}
$factorial
}
function Get-Catalan ([int]$Number)
{
if ($Number -eq 0)
{
return 1
}
(Get-Factorial (2 * $Number)) / ((Get-Factorial (1 + $Number)) * (Get-Factorial $Number))
}
}
Process
{
foreach ($number in $InputObject)
{
[PSCustomObject]@{
Number = $number
CatalanNumber = Get-Catalan $number
}
}
}
}
</syntaxhighlight>
Get the first fifteen Catalan numbers as a PSCustomObject:
<syntaxhighlight lang="powershell">
0..14 | Get-CatalanNumber
</syntaxhighlight>
{{Out}}
<pre>
Number CatalanNumber
------ -------------
0 1
1 1
2 2
3 5
4 14
5 42
6 132
7 429
8 1430
9 4862
10 16796
11 58786
12 208012
13 742900
14 2674440
</pre>
To return only the array of Catalan numbers:
<syntaxhighlight lang="powershell">
(0..14 | Get-CatalanNumber).CatalanNumber
</syntaxhighlight>
{{Out}}
<pre>
1
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440
</pre>
=={{header|Prolog}}==
{{Works with|SWI-Prolog}}
<
length(L1, N),
L = [1 | L1],
Line 1,611 ⟶ 5,100:
my_write(N, V) :-
format('~w : ~w~n', [N, V]).</syntaxhighlight>
{{out}}
<pre> ?- catalan(15).
0 : 1
Line 1,638 ⟶ 5,126:
Python will transparently switch to bignum-type integer arithmetic, so the code below works unchanged on computing larger catalan numbers such as cat(50) and beyond.
{{Works with|Python|3}}
<syntaxhighlight lang="python">from math import factorial
import functools
def memoize(func):
cache = {}
def memoized(key):
# Returned, new, memoized version of decorated function
Line 1,654 ⟶ 5,146:
def fact(n):
return factorial(n)
def cat_direct(n):
return fact(2 * n) // fact(n + 1) // fact(n)
@memoize
def catR1(n):
return
)
@memoize
def catR2(n):
return
)
Line 1,673 ⟶ 5,169:
def pr(results):
fmt = '%-10s %-10s %-10s'
print
print
for r in zip(*results):
print
defs = (cat_direct, catR1, catR2)
results = [
pr(results)</
{{out|Sample Output}}
<pre>CAT_DIRECT CATR1 CATR2
========== ========== ==========
Line 1,702 ⟶ 5,196:
2674440 2674440 2674440</pre>
The third definition is directly expressible, as an infinite series, in terms of '''itertools.accumulate''':
<syntaxhighlight lang="python">'''Catalan numbers'''
from itertools import accumulate, chain, count, islice
# catalans3 :: [Int]
def catalans3():
'''Infinite sequence of Catalan numbers
'''
def go(c, n):
return 2 * c * pred(2 * n) // succ(n)
return accumulate(
chain([1], count(1)), go
)
# ------------------------- TEST -------------------------
# main :: IO ()
def main():
'''Catalan numbers, definition 3'''
print("Catalans 1-15:\n")
print(
'\n'.join([
f'{n:>10}' for n
in islice(catalans3(), 15)
])
)
# ----------------------- GENERIC ------------------------
# pred :: Int -> Int
def pred(n):
'''Predecessor function'''
return n - 1
# succ :: Int -> Int
def succ(n):
'''Successor function'''
return 1 + n
# MAIN ---
if __name__ == '__main__':
main()</syntaxhighlight>
{{Out}}
<pre>Catalans 1-15:
1
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440</pre>
=={{header|Quackery}}==
<syntaxhighlight lang="quackery"> [ 1 over times [ over i 1+ + * ] nip ] is 2n!/n! ( n --> n )
[ times [ i 1+ / ] ] is /n! ( n --> n )
[ dup 2n!/n! swap 1+ /n! ] is catalan ( n --> n )
15 times [ i^ dup echo say " : " catalan echo cr ]</syntaxhighlight>
{{out}}
<pre>0 : 1
1 : 1
2 : 2
3 : 5
4 : 14
5 : 42
6 : 132
7 : 429
8 : 1430
9 : 4862
10 : 16796
11 : 58786
12 : 208012
13 : 742900
14 : 2674440
</pre>
=={{header|R}}==
<syntaxhighlight lang="r">catalan <- function(n) choose(2*n, n)/(n + 1)
catalan(0:15)
[1] 1 1 2 5 14 42 132 429 1430
[10] 4862 16796 58786 208012 742900 2674440 9694845</syntaxhighlight>
=={{header|Racket}}==
<syntaxhighlight lang="racket">#lang racket
(require planet2)
; (install "this-and-that") ; uncomment to install
(require memoize/memo)
(define/memo* (catalan m)
(if (= m 0)
1
(for/sum ([i m])
(* (catalan i) (catalan (- m i 1))))))
(map catalan (range 1 15))</syntaxhighlight>
{{out}}
<pre>
'(1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440)
</pre>
=={{header|Raku}}==
(formerly Perl 6)
{{works with|Rakudo|2015.12}}
The recursive formulas are easily written into a constant array, either:
<syntaxhighlight lang="raku" line>constant Catalan = 1, { [+] @_ Z* @_.reverse } ... *;</syntaxhighlight>
or
<syntaxhighlight lang="raku" line>constant Catalan = 1, |[\*] (2, 6 ... *) Z/ 2 .. *;
# In both cases, the sixteen first values can be seen with:
.say for Catalan[^16];</syntaxhighlight>
{{out}}
<pre>1
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440</pre>
=={{header|REXX}}==
===version 1===
All four methods of calculate the Catalan numbers use independent memoization
for the computation of factorials.
In the 1<sup>st</sup> equation, the 2<sup>nd</sup> version's denominator:
:::::::: <big> (n+1)! n! </big>
has been rearranged to:
:::::::: <big> (n+1) * [fact(n) **2] </big>
<syntaxhighlight lang="rexx">/*REXX program calculates and displays Catalan numbers using four different methods. */
parse arg LO HI . /*obtain optional arguments from the CL*/
if LO=='' | LO=="," then do; HI=15; LO=0; end /*No args? Then use a range of 0 ──► 15*/
if HI=='' | HI=="," then HI=LO /*No HI? Then use LO for the default*/
numeric digits max(20, 5*HI) /*this allows gihugic Catalan numbers. */
w=length(HI) /*W: is used for aligning the output. */
call hdr 1A; do j=LO to HI; say ' Catalan' right(j, w)": " Cat1A(j); end
call hdr 1B; do j=LO to HI; say ' Catalan' right(j, w)": " Cat1B(j); end
call hdr 2 ; do j=LO to HI; say ' Catalan' right(j, w)": " Cat2(j) ; end
call hdr 3 ; do j=LO to HI; say ' Catalan' right(j, w)": " Cat3(j) ; end
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
!: arg z; if !.z\==. then return !.z; !=1; do k=2 to z; !=!*k; end; !.z=!; return !
Cat1A: procedure expose !.; parse arg n; return comb(n+n, n) % (n+1)
Cat1B: procedure expose !.; parse arg n; return !(n+n) % ((n+1) * !(n)**2)
Cat3: procedure expose c.; arg n; if c.n==. then c.n=(4*n-2)*cat3(n-1)%(n+1); return c.n
comb: procedure; parse arg x,y; return pFact(x-y+1, x) % pFact(2, y)
hdr: !.=.; c.=.; c.0=1; say; say center('Catalan numbers, method' arg(1),79,'─'); return
pFact: procedure; !=1; do k=arg(1) to arg(2); !=!*k; end; return !
/*──────────────────────────────────────────────────────────────────────────────────────*/
Cat2: procedure expose c.; parse arg n; $=0; if c.n\==. then return c.n
do k=0 for n; $=$ + Cat2(k) * Cat2(n-k-1); end
c.n=$; return $ /*use a memoization technique.*/</syntaxhighlight>
'''output''' when using the input of: <tt> 0 16 </tt>
<pre>
───────────────────────── Catalan numbers, method 1A ──────────────────────────
Catalan 0: 1
Catalan 1: 1
Catalan 2: 2
Catalan 3: 5
Catalan 4: 14
Catalan 5: 42
Catalan 6: 132
Catalan 7: 429
Catalan 8: 1430
Catalan 9: 4862
Catalan 10: 16796
Catalan 11: 58786
Catalan 12: 208012
Catalan 13: 742900
Catalan 14: 2674440
Catalan 15: 9694845
───────────────────────── Catalan numbers, method 1B ──────────────────────────
··· (elided, same as first method) ···
───────────────────────── Catalan numbers, method 2 ──────────────────────────
··· (elided, same as first method) ···
───────────────────────── Catalan numbers, method 3 ──────────────────────────
··· (elided, same as first method) ···
</pre>
'''Timing notes''' of the four methods:
::* For Catalan numbers 1 ──► 200:
::::* method '''1A''' is about '''50''' times slower than method '''3'''
::::* method '''1B''' is about '''100''' times slower than method '''3'''
::::* method '''2''' is about '''85''' times slower than method '''3'''
::* For Catalan numbers 1 ──► 300:
::::* method '''1A''' is about '''100''' times slower than method '''3'''
::::* method '''1B''' is about '''200''' times slower than method '''3'''
::::* method '''2''' is about '''200''' times slower than method '''3'''
Method '''3''' is really quite fast; even in the thousands range, computation time is still quite reasonable.
===version 2===
Implements the 3 methods shown in the task description
<syntaxhighlight lang="rexx">/* REXX ---------------------------------------------------------------
* 01.07.2014 Walter Pachl
*--------------------------------------------------------------------*/
Numeric Digits 1000
Parse Arg m .
If m='' Then m=20
Do i=0 To m
c1.i=c1(i)
End
c2.=1
Do i=1 To m
c2.i=c2(i)
End
c3.=1
Do i=1 To m
im1=i-1
c3.i=2*(2*i-1)*c3.im1/(i+1)
End
l=length(c3.m)
hdr=' n' right('c1.n',l),
right('c2.n',l),
right('c3.n',l)
Say hdr
Do i=0 To m
Say right(i,2) format(c1.i,l),
format(c2.i,l),
format(c3.i,l)
End
Say hdr
Exit
c1: Procedure
Parse Arg n
return fact(2*n)/(fact(n)*fact(n+1))
c2: Procedure Expose c2.
Parse Arg n
res=0
Do i=0 To n-1
nmi=n-i-1
res=res+c2.i*c2.nmi
End
Return res
fact: Procedure
Parse Arg n
f=1
Do i=1 To n
f=f*i
End
Return f</syntaxhighlight>
{{out}}
<pre> n c1.n c2.n c3.n
0 1 1 1
1 1 1 1
2 2 2 2
3 5 5 5
4 14 14 14
5 42 42 42
6 132 132 132
7 429 429 429
8 1430 1430 1430
9 4862 4862 4862
10 16796 16796 16796
11 58786 58786 58786
12 208012 208012 208012
13 742900 742900 742900
14 2674440 2674440 2674440
15 9694845 9694845 9694845
16 35357670 35357670 35357670
17 129644790 129644790 129644790
18 477638700 477638700 477638700
19 1767263190 1767263190 1767263190
20 6564120420 6564120420 6564120420
n c1.n c2.n c3.n</pre>
=={{header|Ring}}==
<syntaxhighlight lang="ring">
for n = 1 to 15
see catalan(n) + nl
next
func catalan n
if n = 0 return 1 ok
cat = 2 * (2 * n - 1) * catalan(n - 1) / (n + 1)
return cat
</syntaxhighlight>
Output:
<pre>
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440
9694845
</pre>
=={{header|RPL}}==
{{works with|Halcyon Calc|4.2.7}}
{| class="wikitable"
! Code
! Comments
|-
|
≪
'''IFERR''' R→B '''THEN END'''
'''IF''' DUP #1 ≠
'''THEN'''
DUP 2 * 1 - 2 * OVER 1 - →CAT * SWAP 1 + /
'''END'''
≫ ''''→CAT'''' STO
|
''( n -- C(n) )''
Ignore the conversion error if n is already a binary integer
C(1) = 1
Divide by (n+1) at the end to stay in the integer world
|}
To speed up execution, additions can be preferred to multiplications by replacing <code>2 *</code> with <code>DUP +</code>.
The following piece of code will deliver what is required:
≪ 1 { } '''DO''' OVER →CAT B→R + SWAP 1 + SWAP '''UNTIL''' OVER 15 > '''END''' ≫ EVAL
{{out}}
<pre>
1: { 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845 }
</pre>
=={{header|Ruby}}==
<syntaxhighlight lang="ruby">def factorial(n)
(1..n).reduce(1, :*)
end
# direct
def catalan_direct(n)
Line 1,720 ⟶ 5,569:
def catalan_rec1(n)
return 1 if n == 0
(0...n
end
def catalan_rec2(n)
return 1 if n == 0
2*(2*n - 1) * catalan_rec2(n-1) / (n+1)
end
Line 1,734 ⟶ 5,583:
include Memoize
Benchmark.bm(
b.report('
b.report('catalan_rec1') {16.times {|n|
b.report('catalan_rec2') {16.times {|n| catalan_rec2(n)} }
b.report('catalan_rec1(memo)'){16.times {|n| catalan_rec1(n)} }
end
puts "\n direct rec1 rec2"
16.times {|n| puts "%2d :%9d%9d%9d" % [n, catalan_direct(n), catalan_rec1(n), catalan_rec2(n)]}</syntaxhighlight>
The output shows the dramatic difference memoizing makes.
<pre>
user system total real
catalan_rec1 6.178000 0.000000 6.178000 ( 6.195141)
catalan_rec2 0.000000 0.000000 0.000000 ( 0.000023)
catalan_rec1(memo) 0.000000 0.000000 0.000000 ( 0.000641)
direct rec1 rec2
0 : 1 1 1
1 : 1 1 1
2 : 2 2 2
3 : 5 5 5
4 : 14 14 14
5 : 42 42 42
6 : 132 132 132
7 : 429 429 429
8 : 1430 1430 1430
9 : 4862 4862 4862
10 : 16796 16796 16796
11 : 58786 58786 58786
12 : 208012 208012 208012
13 : 742900 742900 742900
14 : 2674440 2674440 2674440
15 : 9694845 9694845 9694845
</pre>
=={{header|Rust}}==
<syntaxhighlight lang="rust">fn c_n(n: u64) -> u64 {
match n {
0 => 1,
_ => c_n(n - 1) * 2 * (2 * n - 1) / (n + 1)
}
}
fn main() {
for i in 1..16 {
println!("c_n({}) = {}", i, c_n(i));
}
}</syntaxhighlight>
{{out}}
<pre>c(1) = 1
c(2) = 2
c(3) = 5
c(4) = 14
c(5) = 42
c(6) = 132
c(7) = 429
c(8) = 1430
c(9) = 4862
c(10) = 16796
c(11) = 58786
c(12) = 208012
c(13) = 742900
c(14) = 2674440
c(15) = 9694845</pre>
=={{header|Scala}}==
Simple and straightforward. Noticeably out of steam without memoizing at about 5000.
<syntaxhighlight lang="scala">
object CatalanNumbers {
def main(args: Array[String]): Unit = {
for (n <- 0 to 15) {
println("catalan(" + n + ") = " + catalan(n))
}
}
def catalan(n: BigInt): BigInt = factorial(2 * n) / (factorial(n + 1) * factorial(n))
def factorial(n: BigInt): BigInt = BigInt(1).to(n).foldLeft(BigInt(1))(_ * _)
}
</syntaxhighlight>
{{out}}
<pre>
catalan(0) = 1
catalan(1) = 1
catalan(2) = 2
catalan(3) = 5
catalan(4) = 14
catalan(5) = 42
catalan(6) = 132
catalan(7) = 429
catalan(8) = 1430
catalan(9) = 4862
catalan(10) = 16796
catalan(11) = 58786
catalan(12) = 208012
catalan(13) = 742900
catalan(14) = 2674440
catalan(15) = 9694845
</pre>
=={{header|Scheme}}==
Tail recursive implementation.
<
(let loop ((c 1)(n 0))
(if (not (eqv? n m))
Line 1,779 ⟶ 5,696:
(loop (* (/ (* 2 (- (* 2 (+ n 1)) 1)) (+ (+ n 1) 1)) c) (+ n 1) )))))
(catalan 15)</syntaxhighlight>
{{out}}
1: 1
2: 2
Line 1,796 ⟶ 5,713:
13: 742900
14: 2674440</pre>
=={{header|Seed7}}==
<syntaxhighlight lang="seed7">$ include "seed7_05.s7i";
include "bigint.s7i";
const proc: main is func
local
var bigInteger: n is 0_;
begin
for n range 0_ to 15_ do
writeln((2_ * n) ! n div succ(n));
end for;
end func;</syntaxhighlight>
{{out}}
<pre>
1
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440
9694845
</pre>
=={{header|Sidef}}==
<syntaxhighlight lang="ruby">func f(i) { i==0 ? 1 : (i * f(i-1)) }
func c(n) { f(2*n) / f(n) / f(n+1) }</syntaxhighlight>
With memoization:
<syntaxhighlight lang="ruby">func c(n) is cached {
n == 0 ? 1 : (c(n-1) * (4 * n - 2) / (n + 1))
}</syntaxhighlight>
Calling the function:
<syntaxhighlight lang="ruby">15.times { |i|
say "#{i}\t#{c(i)}"
}</syntaxhighlight>
{{out}}
<pre>
0 1
1 1
2 2
3 5
4 14
5 42
6 132
7 429
8 1430
9 4862
10 16796
11 58786
12 208012
13 742900
14 2674440
</pre>
=={{header|Standard ML}}==
<syntaxhighlight lang="sml">(*
* val catalan : int -> int
* Returns the nth Catalan number.
Line 1,830 ⟶ 5,807:
* 2674440
* 9694845
*)</syntaxhighlight>
=={{header|Stata}}==
<syntaxhighlight lang="stata">clear
set obs 15
gen catalan=1 in 1
replace catalan=catalan[_n-1]*2*(2*_n-3)/_n in 2/l
list, noobs noh</syntaxhighlight>
'''Output'''
<pre> +---------+
| 1 |
| 1 |
| 2 |
| 5 |
| 14 |
|---------|
| 42 |
| 132 |
| 429 |
| 1430 |
| 4862 |
|---------|
| 16796 |
| 58786 |
| 208012 |
| 742900 |
| 2674440 |
+---------+</pre>
=={{header|Swift}}==
{{trans|Rust}}
<syntaxhighlight lang="swift">func catalan(_ n: Int) -> Int {
switch n {
case 0:
return 1
case _:
return catalan(n - 1) * 2 * (2 * n - 1) / (n + 1)
}
}
for i in 1..<16 {
print("catalan(\(i)) => \(catalan(i))")
}
</syntaxhighlight>
{{out}}
<pre>
catalan(1) => 1
catalan(2) => 2
catalan(3) => 5
catalan(4) => 14
catalan(5) => 42
catalan(6) => 132
catalan(7) => 429
catalan(8) => 1430
catalan(9) => 4862
catalan(10) => 16796
catalan(11) => 58786
catalan(12) => 208012
catalan(13) => 742900
catalan(14) => 2674440
catalan(15) => 9694845
</pre>
=={{header|Tcl}}==
<
# Memoization wrapper
Line 1,852 ⟶ 5,891:
$n == 0 ? 1 : 2 * (2*$n - 1) * catalan($n - 1) / ($n + 1)
}}
}</
Demonstration:
<
puts "C_$i = [expr {catalan($i)}]"
}</
{{out}}
<pre>
C_0 = 1
Line 1,884 ⟶ 5,923:
</pre>
== {{header|
{{trans|GW-BASIC}}
<syntaxhighlight lang="javascript">
// Catalan numbers
var c: number[] = [1];
console.log(`${0}\t${c[0]}`);
for (n = 0; n < 15; n++)
c[n + 1] =
for (i = 0; i <= n;
c[n + 1] = c[n + 1] + c[i] *
console.log(`${n + 1}\t${c[n + 1]}`);
}
</syntaxhighlight>
{{out}}
<pre>
0 1
1 1
2 2
3 5
4 14
5 42
6 132
7 429
8 1430
9 4862
10 16796
11 58786
12 208012
13 742900
14 2674440
15 9694845
</pre>
=={{header|Ursala}}==
<syntaxhighlight lang="ursala">#import std
#import nat
Line 1,914 ⟶ 5,964:
#cast %nL
t = catalan* iota 16</
{{out}}
<pre><
1,
Line 1,933 ⟶ 5,983:
2674440,
9694845></pre>
=={{header|Vala}}==
<syntaxhighlight lang="vala">namespace CatalanNumbers {
public class CatalanNumberGenerator {
private static double factorial(double n) {
if (n == 0)
return 1;
return n * factorial(n - 1);
}
public double first_method(double n) {
const double top_multiplier = 2;
return factorial(top_multiplier * n) / (factorial(n + 1) * factorial(n));
}
public double second_method(double n) {
if (n == 0) {
return 1;
}
double sum = 0;
double i = 0;
for (; i <= (n - 1); i++) {
sum += second_method(i) * second_method((n - 1) - i);
}
return sum;
}
public double third_method(double n) {
if (n == 0) {
return ((2 * (2 * n - 1)) / (n + 1)) * third_method(n - 1);
}
void main() {
CatalanNumberGenerator generator = new CatalanNumberGenerator();
DateTime initial_time;
DateTime final_time;
TimeSpan ts;
stdout.printf("Direct Method\n");
stdout.printf(" n%9s\n", "C_n");
stdout.printf("............\n");
initial_time = new DateTime.now();
for (double i = 0; i <= 15; i++) {
stdout.printf("%2s %8s\n", i.to_string(), Math.ceil(generator.first_method(i)).to_string());
final_time = new DateTime.now();
ts = final_time.difference(initial_time);
stdout.printf("............\n");
stdout.printf("Time Elapsed: %s μs\n", ts.to_string());
stdout.printf("\nRecursive Method 1\n");
stdout.printf(" n%9s\n", "C_n");
stdout.printf("............\n");
initial_time = new DateTime.now();
for (double i = 0; i <= 15; i++) {
stdout.printf("%2s %8s\n", i.to_string(), Math.ceil(generator.second_method(i)).to_string());
}
final_time = new DateTime.now();
ts = final_time.difference(initial_time);
stdout.printf("............\n");
stdout.printf("Time Elapsed: %s μs\n", ts.to_string());
stdout.printf("\nRecursive Method 2\n");
stdout.printf(" n%9s\n", "C_n");
stdout.printf("............\n");
initial_time = new DateTime.now();
for (double i = 0; i <= 15; i++) {
stdout.printf("%2s %8s\n", i.to_string(), Math.ceil(generator.third_method(i)).to_string());
}
final_time = new DateTime.now();
ts = final_time.difference(initial_time);
stdout.printf("............\n");
stdout.printf("Time Elapsed: %s μs\n", ts.to_string());
}
}</syntaxhighlight>
{{out}}
<pre>
Direct Method
n C_n
............
0 1
1 1
2 2
3 5
4 14
5 42
6 132
7 429
8 1430
9 4862
10 16796
11 58786
12 208012
13 742900
14 2674440
15 9694845
............
Time Elapsed: 132 μs
Recursive Method 1
............
10 16796
11 58786
12 208012
13 742900
14 2674440
15 9694845
............
Time Elapsed: 130430 μs
Recursive Method 2
n C_n
............
0 1
1 1
2 2
3 5
4 14
5 42
6 132
7 429
8 1430
9 4862
10 16796
11 58786
12 208012
13 742900
14 2674440
15 9694845
............
Time Elapsed: 76 μs
</pre>
=={{header|V (Vlang)}}==
{{trans|Go}}
<syntaxhighlight lang="v (vlang)">import math.big
fn main() {
mut b:= big.zero_int
for n := i64(0); n < 15; n++ {
b = big.integer_from_i64(n)
b = (b*big.two_int).factorial()/(b.factorial()*(b*big.two_int-b).factorial())
println(b/big.integer_from_i64(n+1))
}
}</syntaxhighlight>
{{out}}
<pre>
1
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440
</pre>
=={{header|Wortel}}==
<syntaxhighlight lang="wortel">; the following number expression calculcates the nth Catalan number
#~ddiFSFmSoFSn
; which stands for: dup dup inc fac swap fac mult swap double fac swap divide
; to get the first 15 Catalan numbers we map this function over a list from 0 to 15
!*#~ddiFSFmSoFSn @til 15
; returns [1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674439.9999999995]</syntaxhighlight>
=={{header|Wren}}==
{{libheader|Wren-fmt}}
{{libheader|Wren-math}}
<syntaxhighlight lang="wren">import "./fmt" for Fmt
import "./math" for Int
var catalan = Fn.new { |n|
if (n < 0) Fiber.abort("Argument must be a non-negative integer")
var prod = 1
var i = n + 2
while (i <= n * 2) {
prod = prod * i
i = i + 1
}
return prod / Int.factorial(n)
}
var catalanRec
catalanRec = Fn.new { |n| (n != 0) ? 2 * (2 * n - 1) * catalanRec.call(n - 1) / (n + 1) : 1 }
System.print(" n Catalan number")
System.print("------------------")
for (i in 0..15) System.print("%(Fmt.d(2, i)) %(catalan.call(i))")
System.print("\nand again using a recursive function:\n")
for (i in 0..15) System.print("%(Fmt.d(2, i)) %(catalanRec.call(i))")</syntaxhighlight>
{{out}}
<pre>
n Catalan number
------------------
0 1
1 1
2 2
3 5
4 14
5 42
6 132
7 429
8 1430
9 4862
10 16796
11 58786
12 208012
13 742900
14 2674440
15 9694845
and again using a recursive function:
0 1
1 1
2 2
3 5
4 14
5 42
6 132
7 429
8 1430
9 4862
10 16796
11 58786
12 208012
13 742900
14 2674440
15 9694845
</pre>
=={{header|XLISP}}==
<syntaxhighlight lang="lisp">(defun catalan (n)
(if (= n 0)
1
(* (/ (* 2 (- (* 2 n) 1)) (+ n 1)) (catalan (- n 1))) ) )
(defun range (x y)
(cons x
(if (< x y)
(range (+ x 1) y) ) ) )
(print (mapcar catalan (range 0 14)))</syntaxhighlight>
{{out}}
<pre>(1 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440)</pre>
=={{header|XPL0}}==
<syntaxhighlight lang="xpl0">code CrLf=9, IntOut=11;
int C, N;
[C:= 1;
IntOut(0, C); CrLf(0);
for N:= 1 to 14 do
[C:= C*2*(2*N-1)/(N+1);
IntOut(0, C); CrLf(0);
];
]</syntaxhighlight>
{{out}}
<pre>
1
1
2
5
14
42
132
429
1430
4862
16796
58786
208012
742900
2674440
</pre>
=={{header|zkl}}==
Uses GMP to calculate big factorials.
<syntaxhighlight lang="zkl">var BN=Import("zklBigNum");
fcn catalan(n){
BN(2*n).factorial() / BN(n+1).factorial() / BN(n).factorial();
}
foreach n in (16){
println("%2d --> %,d".fmt(n, catalan(n)));
}
println("%2d --> %,d".fmt(100, catalan(100)));</syntaxhighlight>
And an iterative solution at works up to the limit of 64 bit ints (n=33). Would be 35 but need to avoid factional intermediate results.
<syntaxhighlight lang="zkl">fcn catalan(n){ (1).reduce(n,fcn(p,n){ 2*(2*n-1)*p/(n+1) },1) }</syntaxhighlight>
{{out}}
<pre>
0 --> 1
1 --> 1
2 --> 2
3 --> 5
4 --> 14
5 --> 42
6 --> 132
7 --> 429
8 --> 1,430
9 --> 4,862
10 --> 16,796
11 --> 58,786
12 --> 208,012
13 --> 742,900
14 --> 2,674,440
15 --> 9,694,845
100 --> 896,519,947,090,131,496,687,170,070,074,100,632,420,837,521,538,745,909,320
</pre>
=={{header|ZX Spectrum Basic}}==
{{trans|C}}
<syntaxhighlight lang="zxbasic">10 FOR i=0 TO 15
20 LET n=i: LET m=2*n
30 LET r=1: LET d=m-n
40 IF d>n THEN LET n=d: LET d=m-n
50 IF m<=n THEN GO TO 90
60 LET r=r*m: LET m=m-1
70 IF (d>1) AND NOT FN m(r,d) THEN LET r=r/d: LET d=d-1: GO TO 70
80 GO TO 50
90 PRINT i;TAB 4;r/(1+n)
100 NEXT i
110 STOP
120 DEF FN m(a,b)=a-INT (a/b)*b: REM Modulus function
</syntaxhighlight>
| |||