Catalan numbers/Pascal's triangle

The task is to print out the first 15 Catalan numbers by extracting them from Pascal's triangle, see Catalan Numbers and the Pascal Triangle.

Ada

Uses package Pascal from the Pascal triangle solution[[1]]

<lang Ada>with Ada.Text_IO, Pascal;

procedure Catalan is

  Last: Positive := 15;   
  Row: Pascal.Row := Pascal.First_Row(2*Last+1);

begin

  for I in 1 .. Last loop
     Row := Pascal.Next_Row(Row);
     Row := Pascal.Next_Row(Row);
     Ada.Text_IO.Put(Integer'Image(Row(I+1)-Row(I+2)));
  end loop;

end Catalan;</lang>

Output:

1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845

AutoHotkey

Works with: AutoHotkey_L

<lang AutoHotkey>/* Generate Catalan Numbers // // smgs: 20th Feb, 2014

/

Array := [], Array[2,1] := Array[2,2] := 1 ; Array inititated and 2nd row of pascal's triangle assigned INI := 3 ; starts with calculating the 3rd row and as such the value Loop, 31 ; every odd row is taken for calculating catalan number as such to obtain 15 we need 2n+1 { if ( A_index > 2 ) { Loop, % A_INDEX { old := ini-1, index := A_index, index_1 := A_index + 1 Array[ini, index_1] := Array[old, index] + Array[old, index_1] Array[ini, 1] := Array[ini, ini] := 1 line .= Array[ini, A_index] " " } ;~ MsgBox % line ; gives rows of pascal's triangle ; calculating every odd row starting from 1st so as to obtain catalan's numbers if ( mod(ini,2) != 0) { StringSplit, res, line, %A_Space% ans := res0//2, ans_1 := ans++ result := result . res%ans_1% - res%ans% " " } line := ini++ } } MsgBox % result</lang>

Produces:

1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845

C++

<lang cpp>// Generate Catalan Numbers // // Nigel Galloway: June 9th., 2012 //

include <iostream>

int main() {

 const int N = 15;
 int t[N+2] = {0,1};
 for(int i = 1; i<=N; i++){
   for(int j = i; j>1; j--) t[j] = t[j] + t[j-1];
   t[i+1] = t[i];
   for(int j = i+1; j>1; j--) t[j] = t[j] + t[j-1];
   std::cout << t[i+1] - t[i] << " ";
 }
 return 0;

}</lang>

Produces:

1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845

D

Translation of: C++

<lang d>void main() {

   import std.stdio;

   enum uint N = 15;
   uint[N + 2] t;
   t[1] = 1;

   foreach (immutable i; 1 .. N + 1) {
       foreach_reverse (immutable j; 2 .. i + 1)
           t[j] += t[j - 1];
       t[i + 1] = t[i];
       foreach_reverse (immutable j; 2 .. i + 2)
           t[j] += t[j - 1];
       write(t[i + 1] - t[i], ' ');
   }

}</lang>

Output:

1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845

Icon and Unicon

The following works in both languages. It avoids computing elements in Pascal's triangle that aren't used.

<lang unicon>link math

procedure main(A)

   limit := (integer(A[1])|15)+1
   every write(right(binocoef(i := 2*seq(0)\limit,i/2)-binocoef(i,i/2+1),30))

end</lang>

Sample run:

J

<lang j> Catalan=. }:@:(}.@:((<0 1)&|:) - }:@:((<0 1)&|:@:(2&|.)))@:(i. +/\@]^:[ #&1)@:(2&+)</lang>

Example use:

<lang j> Catalan 15 1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845

</lang>

A structured derivation of Catalan follows:

<lang j> o=. @: NB. Composition of verbs (functions)

  ( PascalTriangle=. i. ((+/\@]^:[)) #&1 ) 5

1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70

  ( MiddleDiagonal=. (<0 1)&|: )               o PascalTriangle 5

1 2 6 20 70

  ( AdjacentLeft=.   MiddleDiagonal o (2&|.) ) o PascalTriangle 5

1 4 15 1 5

  ( Catalan=. }: o (}. o MiddleDiagonal - }: o AdjacentLeft) o PascalTriangle o (2&+) f. ) 5

1 2 5 14 42

  Catalan

}:@:(}.@:((<0 1)&|:) - }:@:((<0 1)&|:@:(2&|.)))@:(i. +/\@]^:[ #&1)@:(2&+)</lang>

Mathematica

This builds the entire Pascal triangle that's needed and holds it in memory. Very inefficienct, but seems to be what is asked in the problem. <lang Mathematica>nextrow[lastrow_] := Module[{output},

 output = ConstantArray[1, Length[lastrow] + 1];
 Do[
  outputi + 1 = lastrowi + lastrowi + 1;
  , {i, 1, Length[lastrow] - 1}];
 output
 ]

pascaltriangle[size_] := NestList[nextrow, {1}, size] catalannumbers[length_] := Module[{output, basetriangle},

 basetriangle = pascaltriangle[2 length];
 list1 = basetriangle# *2 + 1, # + 1 & /@ Range[length];
 list2 = basetriangle# *2 + 1, # + 2 & /@ Range[length];
 list1 - list2
 ]

(* testing *) catalannumbers[15]</lang>

Output:

{1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845}

MATLAB / Octave

<lang MATLAB>p = pascal(17); diag(p(2:end-1,2:end-1))-diag(p,2)</lang> Output:

PARI/GP

<lang parigp>vector(15,n,binomial(2*n,n)-binomial(2*n,n+1))</lang>

Output:

%1 = [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845]

Perl

Translation of: C++

<lang Perl>use constant N => 15; my @t = (0, 1); for(my $i = 1; $i <= N; $i++) {

   for(my $j = $i; $j > 1; $j--) { $t[$j] += $t[$j-1] }
   $t[$i+1] = $t[$i];
   for(my $j = $i+1; $j>1; $j--) { $t[$j] += $t[$j-1] }
   print $t[$i+1] - $t[$i], " ";

}</lang>

Output:

1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845

Perl 6

<lang perl6>constant @pascal = [1], -> @p { [0, @p Z+ @p, 0] } ... *;

constant @catalan = gather for 2, 4 ... * -> $ix {

   my @row := @pascal[$ix];
   my $mid = +@row div 2;
   take [-] @row[$mid, $mid+1]

}

.say for @catalan[^20];</lang>

Output:

Python

Translation of: C++

<lang python>>>> n = 15 >>> t = [0] * (n + 2) >>> t[1] = 1 >>> for i in range(1, n + 1): for j in range(i, 1, -1): t[j] += t[j - 1] t[i + 1] = t[i] for j in range(i + 1, 1, -1): t[j] += t[j - 1] print(t[i+1] - t[i], end=' ')

1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845 >>> </lang>

Works with: Python version 2.7

<lang python>def catalan_number(n):

   nm = dm = 1
   for k in range(2, n+1):
     nm, dm = ( nm*(n+k), dm*k )
   return nm/dm

print [catalan_number(n) for n in range(1, 16)]

[1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845]</lang>

Racket

lang racket

(define (next-half-row r)

 (define r1 (for/list ([x r] [y (cdr r)]) (+ x y)))
 `(,(* 2 (car r1)) ,@(for/list ([x r1] [y (cdr r1)]) (+ x y)) 1 0))

(let loop ([n 15] [r '(1 0)])

 (cons (- (car r) (cadr r))
       (if (zero? n) '() (loop (sub1 n) (next-half-row r)))))

</lang>

REXX

<lang rexx>/*REXX program obtains Catalan numbers from Pascal's triangle. */ numeric digits 200 /*might have large Catalan nums. */ parse arg N .; if N== then N=15 /*Any args? No, then use default*/ @.=0; @.1=1 /*stem array default, 1st value. */

 do i=1  for N;                         ip=i+1
                do j=i  by -1  for N;   jm=j-1;   @.j=@.j+@.jm;  end /*j*
 @.ip=@.i;      do k=ip by -1  for N;   km=k-1;   @.k=@.k+@.km;  end /*k*
 say @.ip-@.i
 end   /*i*
                                      /*stick a fork in it, we're done.*/</lang>

output when using the default input:

Run BASIC

<lang runbasic>n = 15 dim t(n+2) t(1) = 1 for i = 1 to n

 for  j = i to 1 step -1  : t(j) = t(j) + t(j-1): next j
 t(i+1) = t(i)
 for  j = i+1 to 1 step -1: t(j) = t(j) + t(j-1 : next j

print t(i+1) - t(i);" "; next i</lang>

Output:

1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845

Ruby

<lang tcl>def catalan(num)

 t = [0, 1] #grows as needed
 1.upto(num).map do |i|
   i.downto(1){|j| t[j] += t[j-1]}
   t[i+1] = t[i]
   (i+1).downto(1) {|j| t[j] += t[j-1]}
   t[i+1] - t[i]
 end

end

puts catalan(15).join(", ")</lang>

Output:

1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845

Seed7

<lang seed7>$ include "seed7_05.s7i";

const proc: main is func

 local
   const integer: N is 15;
   var array integer: t is [] (1) & N times 0;
   var integer: i is 0;
   var integer: j is 0;
 begin
   for i range 1 to N do
     for j range i downto 2 do
       t[j] +:= t[j - 1];
     end for;
     t[i + 1] := t[i];
     for j range i + 1 downto 2 do
       t[j] +:= t[j - 1];
     end for;
     write(t[i + 1] - t[i] <& " ");
   end for;
   writeln;
 end func;</lang>

Output:

1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845

Tcl

<lang tcl>proc catalan n {

   set result {}
   array set t {0 0 1 1}
   for {set i 1} {[set k $i] <= $n} {incr i} {

for {set j $i} {$j > 1} {} {incr t($j) $t([incr j -1])} set t([incr k]) $t($i) for {set j $k} {$j > 1} {} {incr t($j) $t([incr j -1])} lappend result [expr {$t($k) - $t($i)}]

   }
   return $result

}

puts [catalan 15]</lang>

Output:

1 2 5 14 42 132 429 1430 4862 16796 58786 208012 742900 2674440 9694845

zkl

Translation of: PARI/GP

using binomial coefficients.

<lang zkl>fcn binomial(n,k){ (1).reduce(k,fcn(p,i,n){ p*(n-i+1)/i },1,n) } (1).pump(15,List,fcn(n){ binomial(2*n,n)-binomial(2*n,n+1) })</lang>

Output:

L(1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,2674440,9694845)