# Bell numbers

Bell numbers
You are encouraged to solve this task according to the task description, using any language you may know.

Bell or exponential numbers are enumerations of the number of different ways to partition a set that has exactly n elements. Each element of the sequence Bn is the number of partitions of a set of size n where order of the elements and order of the partitions are non-significant. E.G.: {a b} is the same as {b a} and {a} {b} is the same as {b} {a}.

So
B0 = 1 trivially. There is only one way to partition a set with zero elements. { }
B1 = 1 There is only one way to partition a set with one element. {a}
B2 = 2 Two elements may be partitioned in two ways. {a} {b}, {a b}
B3 = 5 Three elements may be partitioned in five ways {a} {b} {c}, {a b} {c}, {a} {b c}, {a c} {b}, {a b c}
and so on.

A simple way to find the Bell numbers is construct a Bell triangle, also known as an Aitken's array or Peirce triangle, and read off the numbers in the first column of each row. There are other generating algorithms though, and you are free to choose the best / most appropriate for your case.

Write a routine (function, generator, whatever) to generate the Bell number sequence and call the routine to show here, on this page at least the first 15 and (if your language supports big Integers) 50th elements of the sequence.

If you do use the Bell triangle method to generate the numbers, also show the first ten rows of the Bell triangle.

## 11l

Translation of: Python
```F bellTriangle(n)
[[BigInt]] tri
L(i) 0 .< n
tri.append([BigInt(0)] * i)
tri[1][0] = 1
L(i) 2 .< n
tri[i][0] = tri[i - 1][i - 2]
L(j) 1 .< i
tri[i][j] = tri[i][j - 1] + tri[i - 1][j - 1]
R tri

V bt = bellTriangle(51)
print(‘First fifteen and fiftieth Bell numbers:’)
L(i) 1..15
print(‘#2: #.’.format(i, bt[i][0]))
print(‘50: ’bt[50][0])
print()
print(‘The first ten rows of Bell's triangle:’)
L(i) 1..10
print(bt[i])```
Output:
```First fifteen and fiftieth Bell numbers:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
50: 10726137154573358400342215518590002633917247281

The first ten rows of Bell's triangle:
[1]
[1, 2]
[2, 3, 5]
[5, 7, 10, 15]
[15, 20, 27, 37, 52]
[52, 67, 87, 114, 151, 203]
[203, 255, 322, 409, 523, 674, 877]
[877, 1080, 1335, 1657, 2066, 2589, 3263, 4140]
[4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147]
[21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975]
```

Works with: GNAT version 8.3.0
```with Ada.Text_IO; use Ada.Text_IO;
procedure Main is
type Bell_Triangle is array(Positive range <>, Positive range <>) of Natural;

procedure Print_Rows (Row : in Positive; Triangle : in Bell_Triangle) is
begin
if Row in Triangle'Range(1) then
for I in Triangle'First(1) .. Row loop
Put_Line (Triangle (I, 1)'Image);
end loop;
end if;
end Print_Rows;

procedure Print_Triangle (Num : in Positive; Triangle : in Bell_Triangle) is
begin
if Num in Triangle'Range then
for I in Triangle'First(1) .. Num loop
for J in Triangle'First(2) .. Num loop
if Triangle (I, J) /= 0 then
Put (Triangle (I, J)'Image);
end if;
end loop;
New_Line;
end loop;
end if;
end Print_Triangle;

procedure Bell_Numbers is
Triangle : Bell_Triangle(1..15, 1..15) := (Others => (Others => 0));
Temp     : Positive := 1;
begin

Triangle (1, 1) := 1;

for I in Triangle'First(1) + 1 .. Triangle'Last(1) loop
Triangle (I, 1) := Temp;

for J in Triangle'First(2) .. Triangle'Last(2) - 1 loop
if Triangle (I - 1, J) /= 0 then
Triangle (I, J + 1) := Triangle (I, J) + Triangle (I - 1, J);
else
Temp := Triangle (I, J);
exit;
end if;
end loop;
end loop;

Put_Line ("First 15 Bell numbers:");
Print_Rows (15, Triangle);

New_Line;

Put_Line ("First 10 rows of the Bell triangle:");
Print_Triangle (10, Triangle);
end Bell_Numbers;
begin
Bell_Numbers;
end Main;
```
Output:
```First 15 Bell numbers:
1
1
2
5
15
52
203
877
4140
21147
115975
678570
4213597
27644437
190899322

First 10 rows of the Bell triangle:
1
1 2
2 3 5
5 7 10 15
15 20 27 37 52
52 67 87 114 151 203
203 255 322 409 523 674 877
877 1080 1335 1657 2066 2589 3263 4140
4140 5017 6097 7432 9089 11155 13744 17007 21147
21147 25287 30304 36401 43833 52922 64077 77821 94828 115975
```

## ALGOL 68

Translation of: Delphi
Works with: ALGOL 68G version Any - tested with release 2.8.3.win32

Uses Algol 68G's LONG LONG INT to calculate the numbers up to 50. Calculates the numbers using the triangle algorithm but without storing the triangle as a whole - each line of the triangle replaces the previous one.

```BEGIN # show some Bell numbers #
PROC show bell = ( INT n, LONG LONG INT bell number )VOID:
print( ( whole( n, -2 ), ": ", whole( bell number, 0 ), newline ) );
INT max bell = 50;
[ 0 : max bell - 2 ]LONG LONG INT a; FOR i TO UPB a DO a[ i ] := 0 OD;
a[ 0 ] := 1;
show bell( 1, a[ 0 ] );
FOR n FROM 0 TO UPB a DO
# replace a with the next line of the triangle #
a[ n ] := a[ 0 ];
FOR j FROM n BY -1 TO 1 DO
a[ j - 1 ] +:= a[ j ]
OD;
IF   n < 14       THEN
show bell( n + 2, a[ 0 ] )
ELIF n = UPB a THEN
print( ( "...", newline ) );
show bell( n + 2, a[ 0 ] )
FI
OD
END```
Output:
``` 1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
...
50: 10726137154573358400342215518590002633917247281
```

## ALGOL-M

```begin
integer function index(row, col);
integer row, col;
index := row * (row-1)/ 2 + col;

integer ROWS; ROWS := 15;
begin
decimal(11) array bell[0:ROWS*(ROWS+1)/2];
integer i, j;

bell[index(1, 0)] := 1.;
for i := 2 step 1 until ROWS do
begin
bell[index(i, 0)] := bell[index(i-1, i-2)];
for j := 1 step 1 until i-1 do
bell[index(i,j)] := bell[index(i,j-1)] + bell[index(i-1,j-1)];
end;

write("First fifteen Bell numbers:");
for i := 1 step 1 until ROWS do
begin
write(i);
writeon(": ");
writeon(bell[index(i,0)]);
end;

write("");
write("First ten rows of Bell's triangle:");
for i := 1 step 1 until 10 do
begin
write("");
for j := 0 step 1 until i-1 do
writeon(bell[index(i,j)]);
end;
end;
end```
Output:
```First fifteen Bell numbers:
1:  1.0
2:  1.0
3:  2.0
4:  5.0
5:  15.0
6:  52.0
7:  203.0
8:  877.0
9:  4140.0
10:  21147.0
11:  115975.0
12:  678570.0
13:  4213597.0
14:  27644437.0
15:  190899322.0

First ten rows of Bell's triangle:
1.0
1.0 2.0
2.0 3.0 5.0
5.0 7.0 10.0 15.0
15.0 20.0 27.0 37.0 52.0
52.0 67.0 87.0 114.0 151.0 203.0
203.0 255.0 322.0 409.0 523.0 674.0 877.0
877.0 1080.0 1335.0 1657.0 2066.0 2589.0 3263.0 4140.0
4140.0 5017.0 6097.0 7432.0 9089.0 11155.0 13744.0 17007.0 21147.0
21147.0 25287.0 30304.0 36401.0 43833.0 52922.0 64077.0 77821.0 94828.0 115975.0```

## APL

Works with: Dyalog APL
```bell←{
tr←↑(⊢,(⊂⊃∘⌽+0,+\)∘⊃∘⌽)⍣14⊢,⊂,1
⎕←'First 15 Bell numbers:'
⎕←tr[;1]
⎕←'First 10 rows of Bell''s triangle:'
⎕←tr[⍳10;⍳10]
}
```
Output:
```First 15 Bell numbers:
1 1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 190899322
First 10 rows of Bell's triangle:
1     0     0     0     0     0     0     0     0      0
1     2     0     0     0     0     0     0     0      0
2     3     5     0     0     0     0     0     0      0
5     7    10    15     0     0     0     0     0      0
15    20    27    37    52     0     0     0     0      0
52    67    87   114   151   203     0     0     0      0
203   255   322   409   523   674   877     0     0      0
877  1080  1335  1657  2066  2589  3263  4140     0      0
4140  5017  6097  7432  9089 11155 13744 17007 21147      0
21147 25287 30304 36401 43833 52922 64077 77821 94828 115975```

## Arturo

Translation of: D
```bellTriangle: function[n][
tri: map 0..n-1 'x [ map 0..n 'y -> 0 ]
set get tri 1 0 1
loop 2..n-1 'i [
set get tri i 0  get (get tri i-1) i-2
loop 1..i-1 'j [
set get tri i j (get (get tri i) j-1) + ( get (get tri i-1) j-1)
]
]
return tri
]
bt: bellTriangle 51

loop 1..15 'x ->
print [x "=>" first bt\[x]]
print ["50 =>" first last bt]
print ""
print "The first ten rows of Bell's triangle:"

loop 1..10 'i ->
print filter bt\[i] => zero?
```
Output:
```1 => 1
2 => 1
3 => 2
4 => 5
5 => 15
6 => 52
7 => 203
8 => 877
9 => 4140
10 => 21147
11 => 115975
12 => 678570
13 => 4213597
14 => 27644437
15 => 190899322
50 => 10726137154573358400342215518590002633917247281

The first ten rows of Bell's triangle:
1
1 2
2 3 5
5 7 10 15
15 20 27 37 52
52 67 87 114 151 203
203 255 322 409 523 674 877
877 1080 1335 1657 2066 2589 3263 4140
4140 5017 6097 7432 9089 11155 13744 17007 21147
21147 25287 30304 36401 43833 52922 64077 77821 94828 115975 ```

## AutoHotkey

```;-----------------------------------
Bell_triangle(maxRows){
row := 1, col := 1, Arr := []
Arr[row, col] := 1
while (Arr.Count() < maxRows){
row++
max := Arr[row-1].Count()
Loop % max{
if (col := A_Index) =1
Arr[row, col] := Arr[row-1, max]
Arr[row, col+1] := Arr[row, col] + Arr[row-1, col]
}
}
return Arr
}
;-----------------------------------
Show_Bell_Number(Arr){
for i, obj in Arr{
res .= obj.1 "`n"
}
return Trim(res, "`n")
}
;-----------------------------------
Show_Bell_triangle(Arr){
for i, obj in Arr{
for j, v in obj
res .= v ", "
res := Trim(res, ", ") . "`n"
}
return Trim(res, "`n")
}
;-----------------------------------
```
Examples:
```MsgBox % Show_Bell_Number(Bell_triangle(15))
MsgBox % Show_Bell_triangle(Bell_triangle(10))
return
```
Output:
```1
1
2
5
15
52
203
877
4140
21147
115975
678570
4213597
27644437
190899322
---------------------------
1
1, 2
2, 3, 5
5, 7, 10, 15
15, 20, 27, 37, 52
52, 67, 87, 114, 151, 203
203, 255, 322, 409, 523, 674, 877
877, 1080, 1335, 1657, 2066, 2589, 3263, 4140
4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147
21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975```

## BASIC

### ANSI BASIC

Translation of: QuickBASIC
Works with: Decimal BASIC
```100 REM Bell numbers
110 LET MaxN = 14
120 OPTION BASE 0
130 DIM A(13) ! i.e. DIM A(MaxN - 1), ANSI BASIC does not allow expressions in the bound arguments.
140 FOR I = 0 TO MaxN - 1
150    LET A(I) = 0
160 NEXT I
170 LET N = 0
180 LET A(0) = 1
190 PRINT USING "B(##) = #########": N, A(0)
200 DO WHILE N < MaxN
210    LET A(N) = A(0)
220    FOR J = N TO 1 STEP -1
230       LET A(J - 1) = A(J - 1) + A(J)
240    NEXT J
250    LET N = N + 1
260    PRINT USING "B(##) = #########": N, A(0)
270 LOOP
280 END
```
Output:
```B( 0) =         1
B( 1) =         1
B( 2) =         2
B( 3) =         5
B( 4) =        15
B( 5) =        52
B( 6) =       203
B( 7) =       877
B( 8) =      4140
B( 9) =     21147
B(10) =    115975
B(11) =    678570
B(12) =   4213597
B(13) =  27644437
B(14) = 190899322
```

### Applesoft BASIC

Translation of: C
``` 100  LET ROWS = 15
110  LET M\$ =  CHR\$ (13)
120  LET N = ROWS: GOSUB 500"BELLTRIANGLE"
130  PRINT "FIRST FIFTEEN BELL NUMBERS:"
140  FOR I = 1 TO ROWS
150      LET BR = I:BC = 0: GOSUB 350"GETBELL"
160      HTAB T * 13 + 1
170      PRINT  RIGHT\$ (" " +  STR\$ (I),2)": "BV; MID\$ (M\$,1,T = 2);
180      LET T = T + 1 - (T = 2) * 3
190  NEXT I
200  PRINT M\$"THE FIRST TEN ROWS OF BELL'S TRIANGLE:";
210  FOR I = 1 TO 10
220      LET BR = I:BC = 0: GOSUB 350"GETBELL"
230      PRINT M\$BV;
240      FOR J = 1 TO I - 1
250          IF I - 1 >  = J THEN BR = I:BC = J: GOSUB 350"GETBELL": PRINT " "BV;
260  NEXT J,I
270  END

300  LET BI = BR * (BR - 1) / 2 + BC
310  RETURN

350  GOSUB 300"BELLINDEX"
360  LET BV = TRI(BI)
370  RETURN

400  GOSUB 300"BELLINDEX"
410  LET TRI(BI) = BV
420  RETURN

500  DIM TRI(N * (N + 1) / 2)
510  LET BR = 1:BC = 0:BV = 1: GOSUB 400"SETBELL"
520  FOR I = 2 TO N
530      LET BR = I - 1:BC = I - 2: GOSUB 350"GETBELL"
540      LET BR = I:BC = 0: GOSUB 400"SETBELL"
550      FOR J = 1 TO I - 1
560          LET BR = I:BC = J - 1: GOSUB 350"GETBELL":V = BV
570          LET BR = I - 1:BC = J - 1: GOSUB 350"GETBELL"
580          LET BR = I:BC = J:BV = V + BV: GOSUB 400"SETBELL"
590  NEXT J,I
600  RETURN```

### ASIC

Translation of: Delphi

Compile with the Extended math option.

```REM Bell numbers
DIM A&(13)
FOR I = 0 TO 13
A&(I) = 0
NEXT I
N = 0
A&(0) = 1
GOSUB DisplayRow:
WHILE N <= 13
A&(N) = A&(0)
J = N
WHILE J >= 1
JM1 = J - 1
A&(JM1) = A&(JM1) + A&(J)
J = J - 1
WEND
N = N + 1
GOSUB DisplayRow:
WEND
END

DisplayRow:
PRINT "B(";
SN\$ = STR\$(N)
SN\$ = RIGHT\$(SN\$, 2)
PRINT SN\$;
PRINT ") =";
PRINT A&(0)
RETURN
```
Output:
```B( 0) =           1
B( 1) =           1
B( 2) =           2
B( 3) =           5
B( 4) =          15
B( 5) =          52
B( 6) =         203
B( 7) =         877
B( 8) =        4140
B( 9) =       21147
B(10) =      115975
B(11) =      678570
B(12) =     4213597
B(13) =    27644437
B(14) =   190899322
```

### Chipmunk Basic

Works with: Chipmunk Basic version 3.6.4
Translation of: ASIC
```100 cls
110 dim a(13)
120 for i = 0 to ubound(a) : a(i) = 0 : next i
130 n = 0
140 a(0) = 1
150 displayrow()
160 while n <= ubound(a)
170  a(n) = a(0)
180  j = n
190  while j >= 1
200   jm1 = j-1
210   a(jm1) = a(jm1)+a(j)
220   j = j-1
230  wend
240  n = n+1
250  displayrow()
260 wend
270 end
280 sub displayrow()
290  print "B(";
300  print right\$(str\$(n),2)") = " a(0)
310 return
```

### FreeBASIC

```#define MAX 21

#macro ncp(n, p)
(fact(n)/(fact(p))/(fact(n-p)))
#endmacro

dim as ulongint fact(0 to MAX), bell(0 to MAX)
dim as uinteger n=0, k

fact(0) = 1
for k=1 to MAX
fact(k) = k*fact(k-1)
next k

bell(n) = 1
print n, bell(n)
for n=0 to MAX-1
for k=0 to n
bell(n+1) += ncp(n, k)*bell(k)
next k
print n+1, bell(n+1)
next n```

### GW-BASIC

Works with: Chipmunk Basic
Works with: PC-BASIC version any
Works with: QBasic
Translation of: Chipmunk Basic
```100 CLS
110 DIM A#(13)
120 FOR I = 0 TO UBOUND(A#) : A#(I) = 0 : NEXT I
130 N = 0
140 A#(0) = 1
150 GOSUB 280
160 WHILE N <= 13
170  A#(N) = A#(0)
180  J = N
190  WHILE J >= 1
200   JM1 = J-1
210   A#(JM1) = A#(JM1)+A#(J)
220   J = J-1
230  WEND
240  N = N+1
250  GOSUB 280
260 WEND
270 END
280 REM Display Row
290 PRINT "B(";
300 PRINT RIGHT\$(STR\$(N),2)") = " A#(0)
310 RETURN
```

### MSX Basic

Translation of: Applesoft BASIC
```100 ROWS = 15
110 M\$ = CHR\$(13)
120 N = ROWS: GOSUB 500
130 PRINT "FIRST FIFTEEN BELL NUMBERS:"
140 FOR I = 1 TO ROWS
150   BR = I: BC = 0: GOSUB 350
160   PRINT RIGHT\$(" " + STR\$(I),2); ": "; BV; MID\$(M\$,1,2)
170   T = T + 1 - (T = 2) * 3
180 NEXT I
190 PRINT
200 PRINT "THE FIRST 10 ROWS OF BELL'S TRIANGLE:";
210 FOR I = 1 TO 10
220   BR = I: BC = 0: GOSUB 350
230   PRINT M\$: PRINT BV;
240   FOR J = 1 TO I - 1
250     IF I - 1 >= J THEN BR = I: BC = J: GOSUB 350: PRINT BV;
260 NEXT J, I
270 END
300 BI = BR * (BR-1) / 2 + BC
310 RETURN
350 GOSUB 300
360 BV = TRI(BI)
370 RETURN
400 GOSUB 300
410 TRI(BI) = BV
420 RETURN
500 DIM TRI(N * (N+1) / 2)
510 BR = 1: BC = 0: BV = 1: GOSUB 400
520 FOR I = 2 TO N
530   BR = I - 1: BC = I - 2: GOSUB 350
540   BR = I: BC = 0: GOSUB 400
550   FOR J = 1 TO I - 1
560     BR = I: BC = J - 1: GOSUB 350: V = BV
570     BR = I - 1: BC = J - 1: GOSUB 350
580     BR = I: BC = J: BV = V + BV: GOSUB 400
590 NEXT J, I
600 RETURN
```

### QuickBASIC

Works with: QBasic version 1.1
Translation of: Delphi
```' Bell numbers
CONST MAXN% = 14
DIM A&(MAXN% - 1)
FOR I% = 0 TO MAXN% - 1
A&(I%) = 0
NEXT I%
N% = 0
A&(0) = 1
PRINT USING "B(##) = #########"; N%; A&(0)
WHILE N% < MAXN%
A&(N%) = A&(0)
FOR J% = N% TO 1 STEP -1
A&(J% - 1) = A&(J% - 1) + A&(J%)
NEXT J%
N% = N% + 1
PRINT USING "B(##) = #########"; N%; A&(0)
WEND
END
```
Output:
```B( 0) =         1
B( 1) =         1
B( 2) =         2
B( 3) =         5
B( 4) =        15
B( 5) =        52
B( 6) =       203
B( 7) =       877
B( 8) =      4140
B( 9) =     21147
B(10) =    115975
B(11) =    678570
B(12) =   4213597
B(13) =  27644437
B(14) = 190899322
```

### RapidQ

Translation of: Delphi
Translation of: QuickBASIC – Translated only display statements, the rest is the same.
```' Bell numbers
CONST MAXN% = 14
DIM A&(MAXN% - 1)
FOR I% = 0 TO MAXN% - 1
A&(I%) = 0
NEXT I%
N% = 0
A&(0) = 1
PRINT FORMAT\$("B(%2d) = %9d", N%, A&(0))
WHILE N% < MAXN%
A&(N%) = A&(0)
FOR J% = N% TO 1 STEP -1
A&(J% - 1) = A&(J% - 1) + A&(J%)
NEXT J%
N% = N% + 1
PRINT FORMAT\$("B(%2d) = %9d", N%, A&(0))
WEND
END
```
Output:
```B( 0) =         1
B( 1) =         1
B( 2) =         2
B( 3) =         5
B( 4) =        15
B( 5) =        52
B( 6) =       203
B( 7) =       877
B( 8) =      4140
B( 9) =     21147
B(10) =    115975
B(11) =    678570
B(12) =   4213597
B(13) =  27644437
B(14) = 190899322
```

### Visual Basic .NET

Translation of: C#
```Imports System.Numerics
Imports System.Runtime.CompilerServices

Module Module1

<Extension()>
Sub Init(Of T)(array As T(), value As T)
If IsNothing(array) Then Return
For i = 0 To array.Length - 1
array(i) = value
Next
End Sub

Function BellTriangle(n As Integer) As BigInteger()()
Dim tri(n - 1)() As BigInteger
For i = 0 To n - 1
Dim temp(i - 1) As BigInteger
tri(i) = temp
tri(i).Init(0)
Next
tri(1)(0) = 1
For i = 2 To n - 1
tri(i)(0) = tri(i - 1)(i - 2)
For j = 1 To i - 1
tri(i)(j) = tri(i)(j - 1) + tri(i - 1)(j - 1)
Next
Next
Return tri
End Function

Sub Main()
Dim bt = BellTriangle(51)
Console.WriteLine("First fifteen Bell numbers:")
For i = 1 To 15
Console.WriteLine("{0,2}: {1}", i, bt(i)(0))
Next
Console.WriteLine("50: {0}", bt(50)(0))
Console.WriteLine()
Console.WriteLine("The first ten rows of Bell's triangle:")
For i = 1 To 10
Dim it = bt(i).GetEnumerator()
Console.Write("[")
If it.MoveNext() Then
Console.Write(it.Current)
End If
While it.MoveNext()
Console.Write(", ")
Console.Write(it.Current)
End While
Console.WriteLine("]")
Next
End Sub

End Module
```
Output:
```First fifteen Bell numbers:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
50: 10726137154573358400342215518590002633917247281

The first ten rows of Bell's triangle:
[1]
[1, 2]
[2, 3, 5]
[5, 7, 10, 15]
[15, 20, 27, 37, 52]
[52, 67, 87, 114, 151, 203]
[203, 255, 322, 409, 523, 674, 877]
[877, 1080, 1335, 1657, 2066, 2589, 3263, 4140]
[4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147]
[21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975]```

## C

Translation of: D
```#include <stdio.h>
#include <stdlib.h>

// row starts with 1; col < row
size_t bellIndex(int row, int col) {
return row * (row - 1) / 2 + col;
}

int getBell(int *bellTri, int row, int col) {
size_t index = bellIndex(row, col);
return bellTri[index];
}

void setBell(int *bellTri, int row, int col, int value) {
size_t index = bellIndex(row, col);
bellTri[index] = value;
}

int *bellTriangle(int n) {
size_t length = n * (n + 1) / 2;
int *tri = calloc(length, sizeof(int));
int i, j;

setBell(tri, 1, 0, 1);
for (i = 2; i <= n; ++i) {
setBell(tri, i, 0, getBell(tri, i - 1, i - 2));
for (j = 1; j < i; ++j) {
int value = getBell(tri, i, j - 1) + getBell(tri, i - 1, j - 1);
setBell(tri, i, j, value);
}
}

return tri;
}

int main() {
const int rows = 15;
int *bt = bellTriangle(rows);
int i, j;

printf("First fifteen Bell numbers:\n");
for (i = 1; i <= rows; ++i) {
printf("%2d: %d\n", i, getBell(bt, i, 0));
}

printf("\nThe first ten rows of Bell's triangle:\n");
for (i = 1; i <= 10; ++i) {
printf("%d", getBell(bt, i, 0));
for (j = 1; j < i; ++j) {
printf(", %d", getBell(bt, i, j));
}
printf("\n");
}

free(bt);
return 0;
}
```
Output:
```First fifteen Bell numbers:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322

The first ten rows of Bell's triangle:
1
1, 2
2, 3, 5
5, 7, 10, 15
15, 20, 27, 37, 52
52, 67, 87, 114, 151, 203
203, 255, 322, 409, 523, 674, 877
877, 1080, 1335, 1657, 2066, 2589, 3263, 4140
4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147
21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975```

## C#

Translation of: D
```using System;
using System.Numerics;

namespace BellNumbers {
public static class Utility {
public static void Init<T>(this T[] array, T value) {
if (null == array) return;
for (int i = 0; i < array.Length; ++i) {
array[i] = value;
}
}
}

class Program {
static BigInteger[][] BellTriangle(int n) {
BigInteger[][] tri = new BigInteger[n][];
for (int i = 0; i < n; ++i) {
tri[i] = new BigInteger[i];
tri[i].Init(BigInteger.Zero);
}
tri[1][0] = 1;
for (int i = 2; i < n; ++i) {
tri[i][0] = tri[i - 1][i - 2];
for (int j = 1; j < i; ++j) {
tri[i][j] = tri[i][j - 1] + tri[i - 1][j - 1];
}
}
return tri;
}

static void Main(string[] args) {
var bt = BellTriangle(51);
Console.WriteLine("First fifteen and fiftieth Bell numbers:");
for (int i = 1; i < 16; ++i) {
Console.WriteLine("{0,2}: {1}", i, bt[i][0]);
}
Console.WriteLine("50: {0}", bt[50][0]);
Console.WriteLine();
Console.WriteLine("The first ten rows of Bell's triangle:");
for (int i = 1; i < 11; ++i) {
//Console.WriteLine(bt[i]);
var it = bt[i].GetEnumerator();
Console.Write("[");
if (it.MoveNext()) {
Console.Write(it.Current);
}
while (it.MoveNext()) {
Console.Write(", ");
Console.Write(it.Current);
}
Console.WriteLine("]");
}
}
}
}
```
Output:
```First fifteen and fiftieth Bell numbers:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
50: 10726137154573358400342215518590002633917247281

The first ten rows of Bell's triangle:
[1]
[1, 2]
[2, 3, 5]
[5, 7, 10, 15]
[15, 20, 27, 37, 52]
[52, 67, 87, 114, 151, 203]
[203, 255, 322, 409, 523, 674, 877]
[877, 1080, 1335, 1657, 2066, 2589, 3263, 4140]
[4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147]
[21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975]```

## C++

Library: Boost

Requires C++14 or later. If HAVE_BOOST is defined, we use the cpp_int class from Boost so we can display the 50th Bell number, as shown in the output section below.

```#include <iostream>
#include <vector>

#ifdef HAVE_BOOST
#include <boost/multiprecision/cpp_int.hpp>
typedef boost::multiprecision::cpp_int integer;
#else
typedef unsigned int integer;
#endif

auto make_bell_triangle(int n) {
std::vector<std::vector<integer>> bell(n);
for (int i = 0; i < n; ++i)
bell[i].assign(i + 1, 0);
bell[0][0] = 1;
for (int i = 1; i < n; ++i) {
std::vector<integer>& row = bell[i];
std::vector<integer>& prev_row = bell[i - 1];
row[0] = prev_row[i - 1];
for (int j = 1; j <= i; ++j)
row[j] = row[j - 1] + prev_row[j - 1];
}
return bell;
}

int main() {
#ifdef HAVE_BOOST
const int size = 50;
#else
const int size = 15;
#endif
auto bell(make_bell_triangle(size));

const int limit = 15;
std::cout << "First " << limit << " Bell numbers:\n";
for (int i = 0; i < limit; ++i)
std::cout << bell[i][0] << '\n';

#ifdef HAVE_BOOST
std::cout << "\n50th Bell number is " << bell[49][0] << "\n\n";
#endif

std::cout << "First 10 rows of the Bell triangle:\n";
for (int i = 0; i < 10; ++i) {
std::cout << bell[i][0];
for (int j = 1; j <= i; ++j)
std::cout << ' ' << bell[i][j];
std::cout << '\n';
}
return 0;
}
```
Output:
```First 15 Bell numbers:
1
1
2
5
15
52
203
877
4140
21147
115975
678570
4213597
27644437
190899322

50th Bell number is 10726137154573358400342215518590002633917247281

First 10 rows of the Bell triangle:
1
1 2
2 3 5
5 7 10 15
15 20 27 37 52
52 67 87 114 151 203
203 255 322 409 523 674 877
877 1080 1335 1657 2066 2589 3263 4140
4140 5017 6097 7432 9089 11155 13744 17007 21147
21147 25287 30304 36401 43833 52922 64077 77821 94828 115975
```

## CLU

```bell = cluster is make, get
rep = array[int]

idx = proc (row, col: int) returns (int)
return (row * (row - 1) / 2 + col)
end idx

get = proc (tri: cvt, row, col: int) returns (int)
return (tri[idx(row, col)])
end get

make = proc (rows: int) returns (cvt)
length: int := rows * (rows+1) / 2
arr: rep := rep\$fill(0, length, 0)

arr[idx(1,0)] := 1
for i: int in int\$from_to(2, rows) do
arr[idx(i,0)] := arr[idx(i-1, i-2)]
for j: int in int\$from_to(1, i-1) do
arr[idx(i,j)] := arr[idx(i,j-1)] + arr[idx(i-1,j-1)]
end
end
return(arr)
end make
end bell

start_up = proc ()
rows = 15

po: stream := stream\$primary_output()
belltri: bell := bell\$make(rows)

stream\$putl(po, "The first 15 Bell numbers are:")
for i: int in int\$from_to(1, rows) do
stream\$putl(po, int\$unparse(i)
|| ": " || int\$unparse(bell\$get(belltri, i, 0)))
end

stream\$putl(po, "\nThe first 10 rows of the Bell triangle:")
for row: int in int\$from_to(1, 10) do
for col: int in int\$from_to(0, row-1) do
stream\$putright(po, int\$unparse(bell\$get(belltri, row, col)), 7)
end
stream\$putc(po, '\n')
end
end start_up```
Output:
```The first 15 Bell numbers are:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322

The first 10 rows of the Bell triangle:
1
1      2
2      3      5
5      7     10     15
15     20     27     37     52
52     67     87    114    151    203
203    255    322    409    523    674    877
877   1080   1335   1657   2066   2589   3263   4140
4140   5017   6097   7432   9089  11155  13744  17007  21147
21147  25287  30304  36401  43833  52922  64077  77821  94828 115975```

## Common Lisp

### via Bell triangle

```;; The triangle is a list of arrays; each array is a
;; triangle's row; the last row is at the head of the list.
(defun grow-triangle (triangle)
(if (null triangle)
'(#(1))
(let* ((last-array (car triangle))
(last-length (length last-array))
(new-array (make-array (1+ last-length)
:element-type 'integer)))
;; copy over the last element of the last array
(setf (aref new-array 0) (aref last-array (1- last-length)))
;; fill in the rest of the array
(loop for i from 0
;; the last index of the new array is the length
;; of the last array, which is 1 unit shorter
for j from 1 upto last-length
for sum = (+ (aref last-array i) (aref new-array i))
do (setf (aref new-array j) sum))
;; return the grown list
(cons new-array triangle))))

(defun make-triangle (num)
(if (<= num 1)
(grow-triangle nil)
(grow-triangle (make-triangle (1- num)))))

(defun bell (num)
(cond ((< num 0) nil)
((= num 0) 1)
(t (aref (first (make-triangle num)) (1- num)))))

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; Printing section
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(defparameter *numbers-to-print*
(append
(loop for i upto 19 collect i)
'(49 50)))

(defun array->list (array)
(loop for i upto (1- (length array))
collect (aref array i)))

(defun print-bell-number (index bell-number)
(format t "B_~d (~:r Bell number) = ~:d~%"
index (1+ index) bell-number))

(defun print-bell-triangle (triangle)
(loop for row in (reverse triangle)
do (format t "~{~d~^, ~}~%" (array->list row))))

;; Final invocation
(loop for n in *numbers-to-print* do
(print-bell-number n (bell n)))

(princ #\newline)

(format t "The first 10 rows of Bell triangle:~%")
(print-bell-triangle (make-triangle 10))
```
Output:
```B_0 (first Bell number) = 1
B_1 (second Bell number) = 1
B_2 (third Bell number) = 2
B_3 (fourth Bell number) = 5
B_4 (fifth Bell number) = 15
B_5 (sixth Bell number) = 52
B_6 (seventh Bell number) = 203
B_7 (eighth Bell number) = 877
B_8 (ninth Bell number) = 4,140
B_9 (tenth Bell number) = 21,147
B_10 (eleventh Bell number) = 115,975
B_11 (twelfth Bell number) = 678,570
B_12 (thirteenth Bell number) = 4,213,597
B_13 (fourteenth Bell number) = 27,644,437
B_14 (fifteenth Bell number) = 190,899,322
B_15 (sixteenth Bell number) = 1,382,958,545
B_16 (seventeenth Bell number) = 10,480,142,147
B_17 (eighteenth Bell number) = 82,864,869,804
B_18 (nineteenth Bell number) = 682,076,806,159
B_19 (twentieth Bell number) = 5,832,742,205,057
B_49 (fiftieth Bell number) = 10,726,137,154,573,358,400,342,215,518,590,002,633,917,247,281
B_50 (fifty-first Bell number) = 185,724,268,771,078,270,438,257,767,181,908,917,499,221,852,770

The first 10 rows of Bell triangle:
1
1, 2
2, 3, 5
5, 7, 10, 15
15, 20, 27, 37, 52
52, 67, 87, 114, 151, 203
203, 255, 322, 409, 523, 674, 877
877, 1080, 1335, 1657, 2066, 2589, 3263, 4140
4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147
21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975```

### via Stirling numbers of the second kind

This solution's algorithm is substantially slower than the algorithm based on the Bell triangle, because of the many nested loops.

```;;; Compute bell numbers analytically

;; Compute the factorial
(defun fact (n)
(cond ((< n 0) nil)
((< n 2) 1)
(t (* n (fact (1- n))))))

;; Compute the binomial coefficient (n choose k)
(defun binomial (n k)
(loop for i from 1 upto k
collect (/ (- (1+ n) i) i) into lst
finally (return (reduce #'* lst))))

;; Compute the Stirling number of the second kind
(defun stirling (n k)
(/
(loop for i upto k summing
(* (expt -1 i) (binomial k i) (expt (- k i) n)))
(fact k)))

;; Compute the Bell number
(defun bell (n)
(loop for k upto n summing (stirling n k)))

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; Printing section
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(defparameter *numbers-to-print*
(append
(loop for i upto 19 collect i)
'(49 50)))

(defun print-bell-number (index bell-number)
(format t "B_~d (~:r Bell number) = ~:d~%"
index (1+ index) bell-number))

;; Final invocation
(loop for n in *numbers-to-print* do
(print-bell-number n (bell n)))
```
Output:
```B_0 (first Bell number) = 1
B_1 (second Bell number) = 1
B_2 (third Bell number) = 2
B_3 (fourth Bell number) = 5
B_4 (fifth Bell number) = 15
B_5 (sixth Bell number) = 52
B_6 (seventh Bell number) = 203
B_7 (eighth Bell number) = 877
B_8 (ninth Bell number) = 4,140
B_9 (tenth Bell number) = 21,147
B_10 (eleventh Bell number) = 115,975
B_11 (twelfth Bell number) = 678,570
B_12 (thirteenth Bell number) = 4,213,597
B_13 (fourteenth Bell number) = 27,644,437
B_14 (fifteenth Bell number) = 190,899,322
B_15 (sixteenth Bell number) = 1,382,958,545
B_16 (seventeenth Bell number) = 10,480,142,147
B_17 (eighteenth Bell number) = 82,864,869,804
B_18 (nineteenth Bell number) = 682,076,806,159
B_19 (twentieth Bell number) = 5,832,742,205,057
B_49 (fiftieth Bell number) = 10,726,137,154,573,358,400,342,215,518,590,002,633,917,247,281
B_50 (fifty-first Bell number) = 185,724,268,771,078,270,438,257,767,181,908,917,499,221,852,770```

## Cowgol

Translation of: C
```include "cowgol.coh";

typedef B is uint32;
typedef I is intptr;

sub bellIndex(row: I, col: I): (addr: I) is
addr := (row * (row - 1) / 2 + col) * @bytesof B;
end sub;

sub getBell(row: I, col: I): (bell: B) is
bell := [LOMEM as [B] + bellIndex(row, col)];
end sub;

sub setBell(row: I, col: I, bell: B) is
[LOMEM as [B] + bellIndex(row, col)] := bell;
end sub;

sub bellTriangle(n: I) is
var length := n * (n + 1) / 2;
var bytes := length * @bytesof B;

if HIMEM - LOMEM < bytes then
print("not enough memory\n");
ExitWithError();
end if;

MemZero(LOMEM, bytes);

setBell(1, 0, 1);
var i: I := 2;
while i <= n loop
setBell(i, 0, getBell(i-1, i-2));
var j: I := 1;
while j < i loop
var value := getBell(i, j-1) + getBell(i-1, j-1);
setBell(i, j, value);
j := j + 1;
end loop;
i := i + 1;
end loop;
end sub;

const ROWS := 15;
bellTriangle(ROWS);
print("First fifteen Bell numbers:\n");
var i: I := 1;
while i <= ROWS loop
print_i32(i as uint32);
print(": ");
print_i32(getBell(i, 0) as uint32);
print_nl();
i := i + 1;
end loop;

print("\nThe first ten rows of Bell's triangle:\n");
i := 1;
while i <= 10 loop
var j: I := 0;
loop
print_i32(getBell(i, j) as uint32);
j := j + 1;
if j == i then break;
else print(", ");
end if;
end loop;
i := i + 1;
print_nl();
end loop;```
Output:
```First fifteen Bell numbers:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322

The first ten rows of Bell's triangle:
1
1, 2
2, 3, 5
5, 7, 10, 15
15, 20, 27, 37, 52
52, 67, 87, 114, 151, 203
203, 255, 322, 409, 523, 674, 877
877, 1080, 1335, 1657, 2066, 2589, 3263, 4140
4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147
21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975```

## D

Translation of: Go
```import std.array : uninitializedArray;
import std.bigint;
import std.stdio : writeln, writefln;

auto bellTriangle(int n) {
auto tri = uninitializedArray!(BigInt[][])(n);
foreach (i; 0..n) {
tri[i] = uninitializedArray!(BigInt[])(i);
tri[i][] = BigInt(0);
}
tri[1][0] = 1;
foreach (i; 2..n) {
tri[i][0] = tri[i - 1][i - 2];
foreach (j; 1..i) {
tri[i][j] = tri[i][j - 1] + tri[i - 1][j - 1];
}
}
return tri;
}

void main() {
auto bt = bellTriangle(51);
writeln("First fifteen and fiftieth Bell numbers:");
foreach (i; 1..16) {
writefln("%2d: %d", i, bt[i][0]);
}
writeln("50: ", bt[50][0]);
writeln;
writeln("The first ten rows of Bell's triangle:");
foreach (i; 1..11) {
writeln(bt[i]);
}
}
```
Output:
```First fifteen and fiftieth Bell numbers:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
50: 10726137154573358400342215518590002633917247281

The first ten rows of Bell's triangle:
[1]
[1, 2]
[2, 3, 5]
[5, 7, 10, 15]
[15, 20, 27, 37, 52]
[52, 67, 87, 114, 151, 203]
[203, 255, 322, 409, 523, 674, 877]
[877, 1080, 1335, 1657, 2066, 2589, 3263, 4140]
[4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147]
[21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975]```

## Delphi

A console application written in Delphi 7. It shows a way of calculating Bell numbers without using a triangle. Output numbering is as in the statement of the task, namely B_0 = 1, B_1 = 1, B_2 = 2, ....

```program BellNumbers;

// For Rosetta Code.
// Delphi console application to display the Bell numbers B_0, ..., B_25.
// Uses signed 64-bit integers, the largest integer type available in Delphi.

{\$APPTYPE CONSOLE}

uses SysUtils; // only for the display

const
MAX_N = 25; // maximum index of Bell number within the limits of int64
var
n : integer; // index of Bell number
j : integer; // loop variable
a : array [0..MAX_N - 1] of int64; // working array to build up B_n

{ Subroutine to display that a[0] is the Bell number B_n }
procedure Display();
begin
WriteLn( SysUtils.Format( 'B_%-2d = %d', [n, a[0]]));
end;

(* Main program *)
begin
n := 0;
a[0] := 1;
Display(); // some programmers would prefer Display;
while (n < MAX_N) do begin // and give begin a line to itself
a[n] := a[0];
for j := n downto 1 do inc( a[j - 1], a[j]);
inc(n);
Display();
end;
end.
```
Output:
```B_0  = 1
B_1  = 1
B_2  = 2
B_3  = 5
B_4  = 15
B_5  = 52
B_6  = 203
B_7  = 877
B_8  = 4140
B_9  = 21147
B_10 = 115975
B_11 = 678570
B_12 = 4213597
B_13 = 27644437
B_14 = 190899322
B_15 = 1382958545
B_16 = 10480142147
B_17 = 82864869804
B_18 = 682076806159
B_19 = 5832742205057
B_20 = 51724158235372
B_21 = 474869816156751
B_22 = 4506715738447323
B_23 = 44152005855084346
B_24 = 445958869294805289
B_25 = 4638590332229999353
```

## EasyLang

Translation of: Julia
```func bell n .
len list[] n
list[1] = 1
for i = 2 to n
for j = 1 to i - 2
list[i - j - 1] += list[i - j]
.
list[i] = list[1] + list[i - 1]
.
return list[n]
.
for i = 1 to 15
print bell i
.```

## Elixir

```defmodule Bell do
def triangle(), do: Stream.iterate([1], fn l -> bell_row l, [List.last l] end)
def numbers(), do: triangle() |> Stream.map(&List.first/1)

defp bell_row([], r), do: Enum.reverse r
defp bell_row([a|a_s], r = [r0|_]), do: bell_row(a_s, [a + r0|r])
end

:io.format "The first 15 bell numbers are ~p~n~n",
[Bell.numbers() |> Enum.take(15)]

IO.puts "The 50th Bell number is #{Bell.numbers() |> Enum.take(50) |> List.last}"
IO.puts ""

IO.puts "THe first 10 rows of Bell's triangle:"
IO.inspect(Bell.triangle() |> Enum.take(10))
```
Output:
```The first 15 bell numbers are [1,1,2,5,15,52,203,877,4140,21147,115975,678570,
4213597,27644437,190899322]

The 50th Bell number is 10726137154573358400342215518590002633917247281

THe first 10 rows of Bell's triangle:
[
[1],
[1, 2],
[2, 3, 5],
[5, 7, 10, 15],
[15, 20, 27, 37, 52],
[52, 67, 87, 114, 151, 203],
[203, 255, 322, 409, 523, 674, 877],
[877, 1080, 1335, 1657, 2066, 2589, 3263, 4140],
[4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147],
[21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975]
]
```

## F#

### The function

```// Generate bell triangle. Nigel Galloway: July 6th., 2019
let bell=Seq.unfold(fun g->Some(g,List.scan(+) (List.last g) g))[1I]
```

```bell|>Seq.take 10|>Seq.iter(printfn "%A")
```
Output:
```[1]
[1; 2]
[2; 3; 5]
[5; 7; 10; 15]
[15; 20; 27; 37; 52]
[52; 67; 87; 114; 151; 203]
[203; 255; 322; 409; 523; 674; 877]
[877; 1080; 1335; 1657; 2066; 2589; 3263; 4140]
[4140; 5017; 6097; 7432; 9089; 11155; 13744; 17007; 21147]
[21147; 25287; 30304; 36401; 43833; 52922; 64077; 77821; 94828; 115975]
```
```bell|>Seq.take 15|>Seq.iter(fun n->printf "%A " (List.head n));printfn ""
```
Output:
```1 1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 190899322
```
```printfn "%A" (Seq.head (Seq.item 49 bell))
```
Output:
```10726137154573358400342215518590002633917247281
```

## Factor

### via Aitken's array

Works with: Factor version 0.98
```USING: formatting io kernel math math.matrices sequences vectors ;

: next-row ( prev -- next )
[ 1 1vector ]
[ dup last [ + ] accumulate swap suffix! ] if-empty ;

: aitken ( n -- seq )
V{ } clone swap [ next-row dup ] replicate nip ;

0 50 aitken col [ 15 head ] [ last ] bi
"First 15 Bell numbers:\n%[%d, %]\n\n50th: %d\n\n" printf
"First 10 rows of the Bell triangle:" print
10 aitken [ "%[%d, %]\n" printf ] each
```
Output:
```First 15 Bell numbers:
{ 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322 }

50th: 10726137154573358400342215518590002633917247281

First 10 rows of the Bell triangle:
{ 1 }
{ 1, 2 }
{ 2, 3, 5 }
{ 5, 7, 10, 15 }
{ 15, 20, 27, 37, 52 }
{ 52, 67, 87, 114, 151, 203 }
{ 203, 255, 322, 409, 523, 674, 877 }
{ 877, 1080, 1335, 1657, 2066, 2589, 3263, 4140 }
{ 4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147 }
{ 21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975 }
```

### via recurrence relation

This solution makes use of a recurrence relation involving binomial coefficients.

Works with: Factor version 0.98
```USING: formatting kernel math math.combinatorics sequences ;

: next-bell ( seq -- n )
dup length 1 - [ swap nCk * ] curry map-index sum ;

: bells ( n -- seq )
V{ 1 } clone swap 1 - [ dup next-bell suffix! ] times ;

50 bells [ 15 head ] [ last ] bi
"First 15 Bell numbers:\n%[%d, %]\n\n50th: %d\n" printf
```
Output:
```First 15 Bell numbers:
{ 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322 }

50th: 10726137154573358400342215518590002633917247281
```

### via Stirling sums

This solution defines Bell numbers in terms of sums of Stirling numbers of the second kind.

Works with: Factor version 0.99 development release 2019-07-10
```USING: formatting kernel math math.extras math.ranges sequences ;

: bell ( m -- n )
[ 1 ] [ dup [1,b] [ stirling ] with map-sum ] if-zero ;

50 [ bell ] { } map-integers [ 15 head ] [ last ] bi
"First 15 Bell numbers:\n%[%d, %]\n\n50th: %d\n" printf
```
Output:

As above.

## FutureBasic

FB does not yet offer native support for Big Ints.

```local fn BellNumbers( limit as long )
long j, n = 1

mda(0) = 1
printf @"%2llu. %19llu", n, mda_integer(0)
while ( n < limit )
mda(n) = mda(0)
for j = n to 1 step -1
mda(j - 1) = mda_integer(j - 1) + mda_integer(j)
next
n++
printf @"%2llu. %19llu", n, mda_integer(0)
wend
end fn

fn BellNumbers( 25 )

HandleEvents```
Output:
``` 1.                   1
2.                   2
3.                   5
4.                  15
5.                  52
6.                 203
7.                 877
8.                4140
9.               21147
10.              115975
11.              678570
12.             4213597
13.            27644437
14.           190899322
15.          1382958545
16.         10480142147
17.         82864869804
18.        682076806159
19.       5832742205057
20.      51724158235372
21.     474869816156751
22.    4506715738447323
23.   44152005855084346
24.  445958869294805289
25. 4638590332229999353
```

## Go

```package main

import (
"fmt"
"math/big"
)

func bellTriangle(n int) [][]*big.Int {
tri := make([][]*big.Int, n)
for i := 0; i < n; i++ {
tri[i] = make([]*big.Int, i)
for j := 0; j < i; j++ {
tri[i][j] = new(big.Int)
}
}
tri[1][0].SetUint64(1)
for i := 2; i < n; i++ {
tri[i][0].Set(tri[i-1][i-2])
for j := 1; j < i; j++ {
}
}
return tri
}

func main() {
bt := bellTriangle(51)
fmt.Println("First fifteen and fiftieth Bell numbers:")
for i := 1; i <= 15; i++ {
fmt.Printf("%2d: %d\n", i, bt[i][0])
}
fmt.Println("50:", bt[50][0])
fmt.Println("\nThe first ten rows of Bell's triangle:")
for i := 1; i <= 10; i++ {
fmt.Println(bt[i])
}
}
```
Output:
```First fifteen and fiftieth Bell numbers:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
50: 10726137154573358400342215518590002633917247281

First ten rows of Bell's triangle:
[1]
[1 2]
[2 3 5]
[5 7 10 15]
[15 20 27 37 52]
[52 67 87 114 151 203]
[203 255 322 409 523 674 877]
[877 1080 1335 1657 2066 2589 3263 4140]
[4140 5017 6097 7432 9089 11155 13744 17007 21147]
[21147 25287 30304 36401 43833 52922 64077 77821 94828 115975]
```

## Groovy

Translation of: Java
```class Bell {
private static class BellTriangle {
private List<Integer> arr

BellTriangle(int n) {
int length = (int) (n * (n + 1) / 2)
arr = new ArrayList<>(length)
for (int i = 0; i < length; ++i) {
}

set(1, 0, 1)
for (int i = 2; i <= n; ++i) {
set(i, 0, get(i - 1, i - 2))
for (int j = 1; j < i; ++j) {
int value = get(i, j - 1) + get(i - 1, j - 1)
set(i, j, value)
}
}
}

private static int index(int row, int col) {
if (row > 0 && col >= 0 && col < row) {
return row * (row - 1) / 2 + col
} else {
throw new IllegalArgumentException()
}
}

int get(int row, int col) {
int i = index(row, col)
return arr.get(i)
}

void set(int row, int col, int value) {
int i = index(row, col)
arr.set(i, value)
}
}

static void main(String[] args) {
final int rows = 15
BellTriangle bt = new BellTriangle(rows)

System.out.println("First fifteen Bell numbers:")
for (int i = 0; i < rows; ++i) {
System.out.printf("%2d: %d\n", i + 1, bt.get(i + 1, 0))
}

for (int i = 1; i <= 10; ++i) {
System.out.print(bt.get(i, 0))
for (int j = 1; j < i; ++j) {
System.out.printf(", %d", bt.get(i, j))
}
System.out.println()
}
}
}
```
Output:
```First fifteen Bell numbers:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
1
1, 2
2, 3, 5
5, 7, 10, 15
15, 20, 27, 37, 52
52, 67, 87, 114, 151, 203
203, 255, 322, 409, 523, 674, 877
877, 1080, 1335, 1657, 2066, 2589, 3263, 4140
4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147
21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975```

```bellTri :: [[Integer]]
bellTri =
let f xs = (last xs, xs)
in map snd (iterate (f . uncurry (scanl (+))) (1, [1]))

bell :: [Integer]
bell = map head bellTri

main :: IO ()
main = do
putStrLn "First 10 rows of Bell's Triangle:"
mapM_ print (take 10 bellTri)
putStrLn "\nFirst 15 Bell numbers:"
mapM_ print (take 15 bell)
putStrLn "\n50th Bell number:"
print (bell !! 49)
```
Output:
```First 10 rows of Bell's Triangle:
[1]
[1,2]
[2,3,5]
[5,7,10,15]
[15,20,27,37,52]
[52,67,87,114,151,203]
[203,255,322,409,523,674,877]
[877,1080,1335,1657,2066,2589,3263,4140]
[4140,5017,6097,7432,9089,11155,13744,17007,21147]
[21147,25287,30304,36401,43833,52922,64077,77821,94828,115975]

First 15 Bell numbers:
1
1
2
5
15
52
203
877
4140
21147
115975
678570
4213597
27644437
190899322

50th Bell number:
10726137154573358400342215518590002633917247281```

And, of course, in terms of Control.Arrow or Control.Applicative, the triangle function could also be written as:

```import Control.Arrow

bellTri :: [[Integer]]
bellTri = map snd (iterate ((last &&& id) . uncurry (scanl (+))) (1,[1]))
```

or:

```import Control.Applicative

bellTri :: [[Integer]]
bellTri = map snd (iterate ((liftA2 (,) last id) . uncurry (scanl (+))) (1,[1]))
```

or, as an applicative without the need for an import:

```bellTri :: [[Integer]]
bellTri = map snd (iterate (((,) . last <*> id) . uncurry (scanl (+))) (1, [1]))
```

## J

```   bell=: ([: +/\ (,~ {:))&.>@:{:

,. bell^:(<5) <1
+--------------+
|1             |
+--------------+
|1 2           |
+--------------+
|2 3 5         |
+--------------+
|5 7 10 15     |
+--------------+
|15 20 27 37 52|
+--------------+

{.&> bell^:(<15) <1
1 1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 190899322

{:>bell^:49<1x
185724268771078270438257767181908917499221852770
```

## Java

Translation of: Kotlin
```import java.util.ArrayList;
import java.util.List;

public class Bell {
private static class BellTriangle {
private List<Integer> arr;

BellTriangle(int n) {
int length = n * (n + 1) / 2;
arr = new ArrayList<>(length);
for (int i = 0; i < length; ++i) {
}

set(1, 0, 1);
for (int i = 2; i <= n; ++i) {
set(i, 0, get(i - 1, i - 2));
for (int j = 1; j < i; ++j) {
int value = get(i, j - 1) + get(i - 1, j - 1);
set(i, j, value);
}
}
}

private int index(int row, int col) {
if (row > 0 && col >= 0 && col < row) {
return row * (row - 1) / 2 + col;
} else {
throw new IllegalArgumentException();
}
}

public int get(int row, int col) {
int i = index(row, col);
return arr.get(i);
}

public void set(int row, int col, int value) {
int i = index(row, col);
arr.set(i, value);
}
}

public static void main(String[] args) {
final int rows = 15;
BellTriangle bt = new BellTriangle(rows);

System.out.println("First fifteen Bell numbers:");
for (int i = 0; i < rows; ++i) {
System.out.printf("%2d: %d\n", i + 1, bt.get(i + 1, 0));
}

for (int i = 1; i <= 10; ++i) {
System.out.print(bt.get(i, 0));
for (int j = 1; j < i; ++j) {
System.out.printf(", %d", bt.get(i, j));
}
System.out.println();
}
}
}
```
Output:
```First fifteen Bell numbers:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
1
1, 2
2, 3, 5
5, 7, 10, 15
15, 20, 27, 37, 52
52, 67, 87, 114, 151, 203
203, 255, 322, 409, 523, 674, 877
877, 1080, 1335, 1657, 2066, 2589, 3263, 4140
4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147
21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975```

## jq

Translation of: Julia
```# nth Bell number
def bell:
. as \$n
| if \$n < 0 then "non-negative integer expected"
elif \$n < 2 then 1
else
reduce range(1; \$n) as \$i ([1];
reduce range(1; \$i) as \$j (.;
.[\$i - \$j] as \$x
| .[\$i - \$j - 1] += \$x )
| .[\$i] = .[0] + .[\$i - 1] )
| .[\$n - 1]
end;

range(1;51) | bell```
Output:

For displaying the results, we will first use gojq, the Go implementation of jq, as it supports unbounded-precision integer arithmetic.

```1
2
5
15
...
37450059502461511196505342096431510120174682
628919796303118415420210454071849537746015761
10726137154573358400342215518590002633917247281
185724268771078270438257767181908917499221852770
```

Using the C-based implementation of jq, the results become inexact from bell(23) onwards:

```[1,1]
[2,2]
...
[21,474869816156751]
[22,4506715738447323]
# inexact
[23,44152005855084344]
[24,445958869294805300]
...
[49,1.0726137154573358e+46]
[50,1.8572426877107823e+47]
```

## Julia

Source: Combinatorics at https://github.com/JuliaMath/Combinatorics.jl/blob/master/src/numbers.jl

```"""
bellnum(n)
Compute the ``n``th Bell number.
"""
function bellnum(n::Integer)
if n < 0
throw(DomainError(n))
elseif n < 2
return 1
end
list = Vector{BigInt}(undef, n)
list[1] = 1
for i = 2:n
for j = 1:i - 2
list[i - j - 1] += list[i - j]
end
list[i] = list[1] + list[i - 1]
end
return list[n]
end

for i in 1:50
println(bellnum(i))
end
```
Output:
```1
2
5
15
52
203
877
4140
21147
115975
678570
4213597
27644437
190899322
1382958545
10480142147
82864869804
682076806159
5832742205057
51724158235372
474869816156751
4506715738447323
44152005855084346
445958869294805289
4638590332229999353
49631246523618756274
545717047936059989389
6160539404599934652455
71339801938860275191172
846749014511809332450147
10293358946226376485095653
128064670049908713818925644
1629595892846007606764728147
21195039388640360462388656799
281600203019560266563340426570
3819714729894818339975525681317
52868366208550447901945575624941
746289892095625330523099540639146
10738823330774692832768857986425209
157450588391204931289324344702531067
2351152507740617628200694077243788988
35742549198872617291353508656626642567
552950118797165484321714693280737767385
8701963427387055089023600531855797148876
139258505266263669602347053993654079693415
2265418219334494002928484444705392276158355
37450059502461511196505342096431510120174682
628919796303118415420210454071849537746015761
10726137154573358400342215518590002633917247281
185724268771078270438257767181908917499221852770
```

## Kotlin

Translation of: C
```class BellTriangle(n: Int) {
private val arr: Array<Int>

init {
val length = n * (n + 1) / 2
arr = Array(length) { 0 }

set(1, 0, 1)
for (i in 2..n) {
set(i, 0, get(i - 1, i - 2))
for (j in 1 until i) {
val value = get(i, j - 1) + get(i - 1, j - 1)
set(i, j, value)
}
}
}

private fun index(row: Int, col: Int): Int {
require(row > 0)
require(col >= 0)
require(col < row)
return row * (row - 1) / 2 + col
}

operator fun get(row: Int, col: Int): Int {
val i = index(row, col)
return arr[i]
}

private operator fun set(row: Int, col: Int, value: Int) {
val i = index(row, col)
arr[i] = value
}
}

fun main() {
val rows = 15
val bt = BellTriangle(rows)

println("First fifteen Bell numbers:")
for (i in 1..rows) {
println("%2d: %d".format(i, bt[i, 0]))
}

for (i in 1..10) {
print("\${bt[i, 0]}")
for (j in 1 until i) {
print(", \${bt[i, j]}")
}
println()
}
}
```
Output:
```First fifteen Bell numbers:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
1
1, 2
2, 3, 5
5, 7, 10, 15
15, 20, 27, 37, 52
52, 67, 87, 114, 151, 203
203, 255, 322, 409, 523, 674, 877
877, 1080, 1335, 1657, 2066, 2589, 3263, 4140
4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147
21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975```

## Little Man Computer

Demonstrates an algorithm that uses a 1-dimensional array and addition. The maximum integer on the LMC is 999, so the maximum Bell number found is B_7 = 877.

In order to handle arrays, a program for the LMC has to modify its own code. This practice is usually frowned on nowadays, but was standard on very early real-life computers, such as EDSAC.

```// Little Man Computer, for Rosetta Code.
// Calculate Bell numbers, using a 1-dimensional array and addition.
//
// After the calculation of B_n (n > 0), the array contains n elements,
// of which B_n is the first. Example to show calculation of B_(n+1):
// After calc. of B_3 = 5, array holds:      5,  3,  2
// Extend array by copying B_3 to high end:  5,  3,  2,  5
// Replace 2 by 5 + 2 = 7:                   5,  3,  7,  5
// Replace 3 by 7 + 3 = 10:                  5, 10,  7,  5
// Replace first 5 by 10 + 5 = 15:          15, 10,  7,  5
// First element of array is now B_4 = 15.

// Initialize; B_0 := 1
LDA c1
STA Bell
LDA c0
STA index
BRA print    // skip increment of index
// Increment index of Bell number
inc_ix   LDA index
STA index
// Here acc = index; print index and Bell number
print    OUT
LDA colon
OTC          // non-standard instruction; cosmetic only
LDA Bell
OUT
LDA index
BRZ inc_ix   // if index = 0, skip rest and loop back
SUB c7       // reached maximum index yet?
BRZ done     // if so, jump to exit
// Manufacture some instructions
LDA lda_0
STA lda_ix
SUB c200     // convert LDA to STA with same address
STA sta_ix
// Copy latest Bell number to end of array
lda_0    LDA Bell     // load Bell number
sta_ix   STA 0        // address was filled in above
// Manufacture more instructions
LDA lda_ix   // load LDA instruction
loop     SUB c401     // convert to ADD with address 1 less
ADD c200     // convert to STA
STA sta_ix_1
// Execute instructions; zero addresses were filled in above
lda_ix   LDA 0        // load element of array
sta_ix_1 STA 0        // update element below
LDA sta_ix_1 // load previous STA instruction
SUB sta_Bell // does it refer to first element of array?
BRZ inc_ix   // yes, loop to inc index and print
LDA lda_ix   // no, repeat with addresses 1 less
SUB c1
STA lda_ix
BRA loop
// Here when done
done     HLT
// Constants
colon    DAT 58
c0       DAT 0
c1       DAT 1
c7       DAT 7        // maximum index
c200     DAT 200
c401     DAT 401
sta_Bell STA Bell     // not executed; used for comparison
// Variables
index    DAT
Bell     DAT
// Rest of array goes here
// end```
Output:
```[formatted manually]
0:1
1:1
2:2
3:5
4:15
5:52
6:203
7:877
```

## Lua

```-- Bell numbers in Lua
-- db 6/11/2020 (to replace missing original)

local function bellTriangle(n)
local tri = { {1} }
for i = 2, n do
tri[i] = { tri[i-1][i-1] }
for j = 2, i do
tri[i][j] = tri[i][j-1] + tri[i-1][j-1]
end
end
return tri
end

local N = 25 -- in lieu of 50, practical limit with double precision floats
local tri = bellTriangle(N)

print("First 15 and "..N.."th Bell numbers:")
for i = 1, 15 do
print(i, tri[i][1])
end
print(N, tri[N][1])

print()

print("First 10 rows of Bell triangle:")
for i = 1, 10 do
print("[ " .. table.concat(tri[i],", ") .. " ]")
end
```
Output:
```First 15 and 25th Bell numbers:
1       1
2       1
3       2
4       5
5       15
6       52
7       203
8       877
9       4140
10      21147
11      115975
12      678570
13      4213597
14      27644437
15      190899322
25      445958869294805289

First 10 rows of Bell triangle:
[ 1 ]
[ 1, 2 ]
[ 2, 3, 5 ]
[ 5, 7, 10, 15 ]
[ 15, 20, 27, 37, 52 ]
[ 52, 67, 87, 114, 151, 203 ]
[ 203, 255, 322, 409, 523, 674, 877 ]
[ 877, 1080, 1335, 1657, 2066, 2589, 3263, 4140 ]
[ 4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147 ]
[ 21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975 ]```

## Maple

```bell1:=proc(n)
option remember;
end:
bell1(0):=1:

bell1(50);
#        185724268771078270438257767181908917499221852770

combinat[bell](50);
#        185724268771078270438257767181908917499221852770

bell1~([\$0..20]);
#  [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570,
#    4213597, 27644437, 190899322, 1382958545, 10480142147,
#    82864869804, 682076806159, 5832742205057, 51724158235372]```

## Mathematica / Wolfram Language

Function definition:

```BellTriangle[n_Integer?Positive] := NestList[Accumulate[# /. {a___, b_} :> {b, a, b}] &, {1}, n - 1];
BellNumber[n_Integer] := BellTriangle[n][[n, 1]];
```

Output:

```In[51]:= Array[BellNumber, 25]

Out[51]= {1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, \
4213597, 27644437, 190899322, 1382958545, 10480142147, 82864869804, \
682076806159, 5832742205057, 51724158235372, 474869816156751, \
4506715738447323, 44152005855084346, 445958869294805289}

In[52]:= BellTriangle[10]

Out[52]= {{1}, {1, 2}, {2, 3, 5}, {5, 7, 10, 15}, {15, 20, 27, 37,
52}, {52, 67, 87, 114, 151, 203}, {203, 255, 322, 409, 523, 674,
877}, {877, 1080, 1335, 1657, 2066, 2589, 3263, 4140}, {4140, 5017,
6097, 7432, 9089, 11155, 13744, 17007, 21147}, {21147, 25287, 30304,
36401, 43833, 52922, 64077, 77821, 94828, 115975}}
```

## Maxima

It exists in Maxima the belln built-in function.

Below is another way

```/* Subfactorial numbers */
subfactorial(n):=block(
subf[0]:1,
subf[n]:n*subf[n-1]+(-1)^n,
subf[n])\$

/* Bell numbers implementation */
my_bell(n):=if n=0 then 1 else block(
makelist((1/((n-1)!))*subfactorial(j)*binomial(n-1,j)*(n-j)^(n-1),j,0,n-1),
apply("+",%%))\$

/* First 50 */
block(
makelist(my_bell(u),u,0,49),
table_form(%%));
```
Output:
```matrix(
[1],
[1],
[2],
[5],
[15],
[52],
[203],
[877],
[4140],
[21147],
[115975],
[678570],
[4213597],
[27644437],
[190899322],
[1382958545],
[10480142147],
[82864869804],
[682076806159],
[5832742205057],
[51724158235372],
[474869816156751],
[4506715738447323],
[44152005855084346],
[445958869294805289],
[4638590332229999353],
[49631246523618756274],
[545717047936059989389],
[6160539404599934652455],
[71339801938860275191172],
[846749014511809332450147],
[10293358946226376485095653],
[128064670049908713818925644],
[1629595892846007606764728147],
[21195039388640360462388656799],
[281600203019560266563340426570],
[3819714729894818339975525681317],
[52868366208550447901945575624941],
[746289892095625330523099540639146],
[10738823330774692832768857986425209],
[157450588391204931289324344702531067],
[2351152507740617628200694077243788988],
[35742549198872617291353508656626642567],
[552950118797165484321714693280737767385],
[8701963427387055089023600531855797148876],
[139258505266263669602347053993654079693415],
[2265418219334494002928484444705392276158355],
[37450059502461511196505342096431510120174682],
[628919796303118415420210454071849537746015761],
[10726137154573358400342215518590002633917247281]
)
```

## Modula-2

Translation of: QuickBASIC
Works with: ADW Modula-2 version any (Compile with the linker option Console Application).
```MODULE BellNumbers;

FROM STextIO IMPORT
WriteLn, WriteString;
FROM SWholeIO IMPORT
WriteInt;

CONST
MaxN = 14;

VAR
A: ARRAY [0 .. MaxN - 1] OF CARDINAL;
I, J, N: CARDINAL;

PROCEDURE DisplayRow(N, BellNum: CARDINAL);
BEGIN
WriteString("B(");
WriteInt(N, 2);
WriteString(") = ");
WriteInt(BellNum, 9);
WriteLn
END DisplayRow;

BEGIN
FOR I := 0 TO MaxN - 1 DO
A[I] := 0
END;
N := 0;
A[0] := 1;
DisplayRow(N, A[0]);
WHILE N < MaxN DO
A[N] := A[0];
FOR J := N TO 1 BY -1 DO
A[J - 1] := A[J - 1] + A[J]
END;
N := N + 1;
DisplayRow(N, A[0])
END
END BellNumbers.
```
Output:
```B( 0) =         1
B( 1) =         1
B( 2) =         2
B( 3) =         5
B( 4) =        15
B( 5) =        52
B( 6) =       203
B( 7) =       877
B( 8) =      4140
B( 9) =     21147
B(10) =    115975
B(11) =    678570
B(12) =   4213597
B(13) =  27644437
B(14) = 190899322
```

## Nim

### Using Recurrence relation

```import math

iterator b(): int =
## Iterator yielding the bell numbers.
var numbers = @[1]
yield 1
var n = 0
while true:
var next = 0
for k in 0..n:
next += binom(n, k) * numbers[k]
yield next
inc n

when isMainModule:

import strformat

const Limit = 25      # Maximum index beyond which an overflow occurs.

echo "Bell numbers from B0 to B25:"
var i = 0
for n in b():
echo fmt"{i:2d}: {n:>20d}"
inc i
if i > Limit:
break
```
Output:
```Bell numbers from B0 to B25:
0:                    1
1:                    1
2:                    2
3:                    5
4:                   15
5:                   52
6:                  203
7:                  877
8:                 4140
9:                21147
10:               115975
11:               678570
12:              4213597
13:             27644437
14:            190899322
15:           1382958545
16:          10480142147
17:          82864869804
18:         682076806159
19:        5832742205057
20:       51724158235372
21:      474869816156751
22:     4506715738447323
23:    44152005855084346
24:   445958869294805289
25:  4638590332229999353```

### Using Bell triangle

```iterator b(): int =
## Iterator yielding the bell numbers.
var row = @[1]
yield 1
yield 1
while true:
var newRow = newSeq[int](row.len + 1)
newRow[0] = row[^1]
for i in 1..newRow.high:
newRow[i] = newRow[i - 1] + row[i - 1]
row = move(newRow)
yield row[^1]   # The last value of the row is one step ahead of the first one.

iterator bellTriangle(): seq[int] =
## Iterator yielding the rows of the Bell triangle.
var row = @[1]
yield row
while true:
var newRow = newSeq[int](row.len + 1)
newRow[0] = row[^1]
for i in 1..newRow.high:
newRow[i] = newRow[i - 1] + row[i - 1]
row = move(newRow)
yield row

when isMainModule:

import strformat
import strutils

const Limit = 25      # Maximum index beyond which an overflow occurs.

echo "Bell numbers from B0 to B25:"
var i = 0
for n in b():
echo fmt"{i:2d}: {n:>20d}"
inc i
if i > Limit:
break

echo "\nFirst ten rows of Bell triangle:"
i = 0
for row in bellTriangle():
inc i
var line = ""
for val in row:
echo line
if i == 10:
break
```
Output:
```Bell numbers from B0 to B25:
0:                    1
1:                    1
2:                    2
3:                    5
4:                   15
5:                   52
6:                  203
7:                  877
8:                 4140
9:                21147
10:               115975
11:               678570
12:              4213597
13:             27644437
14:            190899322
15:           1382958545
16:          10480142147
17:          82864869804
18:         682076806159
19:        5832742205057
20:       51724158235372
21:      474869816156751
22:     4506715738447323
23:    44152005855084346
24:   445958869294805289
25:  4638590332229999353

First ten rows of Bell triangle:
1
1      2
2      3      5
5      7     10     15
15     20     27     37     52
52     67     87    114    151    203
203    255    322    409    523    674    877
877   1080   1335   1657   2066   2589   3263   4140
4140   5017   6097   7432   9089  11155  13744  17007  21147
21147  25287  30304  36401  43833  52922  64077  77821  94828 115975```

## PARIGP

From the code at OEIS A000110,

```genit(maxx=50)={bell=List();
for(n=0,maxx,q=sum(k=0,n,stirling(n,k,2));
listput(bell,q));bell}
END
```

Output: List([1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322, 1382958545, 10480142147, 82864869804, 682076806159, 5832742205057, 51724158235372, 474869816156751, 4506715738447323, 44152005855084346, 445958869294805289, 4638590332229999353, 49631246523618756274, 545717047936059989389, 6160539404599934652455, 71339801938860275191172, 846749014511809332450147, 10293358946226376485095653, 128064670049908713818925644, 1629595892846007606764728147, 21195039388640360462388656799, 281600203019560266563340426570, 3819714729894818339975525681317, 52868366208550447901945575624941, 746289892095625330523099540639146, 10738823330774692832768857986425209, 157450588391204931289324344702531067, 2351152507740617628200694077243788988, 35742549198872617291353508656626642567, 552950118797165484321714693280737767385, 8701963427387055089023600531855797148876, 139258505266263669602347053993654079693415, 2265418219334494002928484444705392276158355, 37450059502461511196505342096431510120174682, 628919796303118415420210454071849537746015761, 10726137154573358400342215518590002633917247281, 185724268771078270438257767181908917499221852770])

## Pascal

Works with: Free Pascal

Using bell's triangle. TIO.RUN up to 5000.See talk for more.

```program BellNumbers;
{\$Ifdef FPC}
{\$optimization on,all}
{\$ElseIf}
{Apptype console}
{\$EndIf}
uses
sysutils,gmp;
var
T0 :TDateTime;
procedure BellNumbersUint64(OnlyBellNumbers:Boolean);
var
BList : array[0..24] of Uint64;
BellNum : Uint64;
BListLenght,i :nativeUInt;
begin
IF OnlyBellNUmbers then
Begin
writeln('Bell triangles ');
writeln('  1 = 1');
end
else
Begin
writeln('Bell numbers');
writeln('  1 = 1');
writeln('  2 = 1');
end;

BList[0]:= 1;
BListLenght := 1;
BellNum := 1;
repeat
//  For i := BListLenght downto 1 do BList[i] := BList[i-1]; or
move(Blist[0],Blist[1],BListLenght*SizeOf(Blist[0]));
BList[0] := BellNum;
For i := 1 to BListLenght do
Begin
BellNum += BList[i];
BList[i] := BellNum;
end;

//  Output
IF OnlyBellNUmbers then
Begin
IF BListLenght<=9 then
Begin
write(BListLenght+1:3,' = ');
For i := 0 to BListLenght do
write(BList[i]:7);
writeln;
end
ELSE
BREAK;
end
else
writeln(BListLenght+2:3,' = ',BellNum);

inc(BListLenght);
until  BListLenght >= 25;
writeln;
end;

procedure BellNumbersMPInteger;
const
MaxIndex = 5000;//must be > 0
var
//MPInteger as alternative to mpz_t -> selfcleaning
BList : array[0..MaxIndex] of MPInteger;
BellNum : MPInteger;
BListLenght,i :nativeUInt;
BellNumStr : AnsiString;
Begin
BellNumStr := '';
z_init(BellNum);
z_ui_pow_ui(BellNum,10,32767);
BListLenght := z_size(BellNum);
writeln('init length ',BListLenght);
For i := 0 to MaxIndex do
Begin
//    z_init2_set(BList[i],BListLenght);
end;
writeln('init length ',z_size(BList[0]));

T0 := now;
BListLenght := 1;
z_set_ui(BList[0],1);
z_set_ui(BellNum,1);
repeat
//Move does not fit moving interfaces //    call    fpc_intf_assign
For i := BListLenght downto 1 do  BList[i] := BList[i-1];
z_set(BList[0],BellNum);
For i := 1 to BListLenght do
Begin
z_set(BList[i],BellNum);
end;
inc(BListLenght);
if (BListLenght+1) MOD 100 = 0 then
Begin
BellNumStr:= z_get_str(10,BellNum);
//z_sizeinbase (BellNum, 10) is not exact :-(
write('Bell(',(IntToStr(BListLenght)):6,') has ',
(IntToStr(Length(BellNumStr))):6,' decimal digits');
writeln(FormatDateTime(' NN:SS.ZZZ',now-T0),'s');
end;
until  BListLenght>=MaxIndex;
BellNumStr:= z_get_str(10,BellNum);
writeln(BListLenght:6,'.th ',Length(BellNumStr):8);

//clean up ;-)
BellNumStr := '';
z_clear(BellNum);
For i := MaxIndex downto 0 do
z_clear(BList[i]);
end;

BEGIN
BellNumbersUint64(True);BellNumbersUint64(False);
BellNumbersMPInteger;
END.
```
Output:
```TIO.RUN
Real time: 22.818 s User time: 22.283 s Sys. time: 0.109 s CPU share: 98.13 %

Bell triangles
1 = 1
2 =       1      2
3 =       2      3      5
4 =       5      7     10     15
5 =      15     20     27     37     52
6 =      52     67     87    114    151    203
7 =     203    255    322    409    523    674    877
8 =     877   1080   1335   1657   2066   2589   3263   4140
9 =    4140   5017   6097   7432   9089  11155  13744  17007  21147
10 =   21147  25287  30304  36401  43833  52922  64077  77821  94828 115975

Bell numbers
1 = 1
2 = 1
3 = 2
4 = 5
5 = 15
6 = 52
7 = 203
8 = 877
9 = 4140
10 = 21147
11 = 115975
12 = 678570
13 = 4213597
14 = 27644437
15 = 190899322
16 = 1382958545
17 = 10480142147
18 = 82864869804
19 = 682076806159
20 = 5832742205057
21 = 51724158235372
22 = 474869816156751
23 = 4506715738447323
24 = 44152005855084346
25 = 445958869294805289
26 = 4638590332229999353

init length 1701
init length 0
Bell(    99) has    115 decimal digits 00:00.001s
Bell(   199) has    275 decimal digits 00:00.005s
Bell(   299) has    453 decimal digits 00:00.013s
Bell(   399) has    643 decimal digits 00:00.022s
Bell(   499) has    842 decimal digits 00:00.035s
Bell(   599) has   1048 decimal digits 00:00.051s
Bell(   699) has   1260 decimal digits 00:00.071s
Bell(   799) has   1478 decimal digits 00:00.098s
Bell(   899) has   1700 decimal digits 00:00.128s
Bell(   999) has   1926 decimal digits 00:00.167s
Bell(  1099) has   2155 decimal digits 00:00.208s
Bell(  1199) has   2388 decimal digits 00:00.256s
Bell(  1299) has   2625 decimal digits 00:00.310s
Bell(  1399) has   2864 decimal digits 00:00.366s
Bell(  1499) has   3105 decimal digits 00:00.440s
Bell(  1599) has   3349 decimal digits 00:00.517s
Bell(  1699) has   3595 decimal digits 00:00.608s
Bell(  1799) has   3844 decimal digits 00:00.711s
Bell(  1899) has   4095 decimal digits 00:00.808s
Bell(  1999) has   4347 decimal digits 00:00.959s
Bell(  2099) has   4601 decimal digits 00:01.189s
Bell(  2199) has   4858 decimal digits 00:01.373s
Bell(  2299) has   5115 decimal digits 00:01.560s
Bell(  2399) has   5375 decimal digits 00:01.816s
Bell(  2499) has   5636 decimal digits 00:02.065s
Bell(  2599) has   5898 decimal digits 00:02.304s
Bell(  2699) has   6162 decimal digits 00:02.669s
Bell(  2799) has   6428 decimal digits 00:02.962s
Bell(  2899) has   6694 decimal digits 00:03.366s
Bell(  2999) has   6962 decimal digits 00:03.720s
Bell(  3099) has   7231 decimal digits 00:04.145s
Bell(  3199) has   7502 decimal digits 00:04.554s
Bell(  3299) has   7773 decimal digits 00:05.198s
Bell(  3399) has   8046 decimal digits 00:05.657s
Bell(  3499) has   8320 decimal digits 00:06.270s
Bell(  3599) has   8595 decimal digits 00:06.804s
Bell(  3699) has   8871 decimal digits 00:07.475s
Bell(  3799) has   9148 decimal digits 00:08.189s
Bell(  3899) has   9426 decimal digits 00:08.773s
Bell(  3999) has   9704 decimal digits 00:09.563s
Bell(  4099) has   9984 decimal digits 00:10.411s
Bell(  4199) has  10265 decimal digits 00:11.301s
Bell(  4299) has  10547 decimal digits 00:12.230s
Bell(  4399) has  10829 decimal digits 00:13.415s
Bell(  4499) has  11112 decimal digits 00:14.830s
Bell(  4599) has  11397 decimal digits 00:16.630s
Bell(  4699) has  11682 decimal digits 00:18.210s
Bell(  4799) has  11968 decimal digits 00:19.964s
Bell(  4899) has  12254 decimal digits 00:21.332s
Bell(  4999) has  12542 decimal digits 00:22.445s
5000.th    12544
```

## Perl

Translation of: Raku
```use strict 'vars';
use warnings;
use feature 'say';
use bigint;

my @b = 1;
my @Aitkens = [1];

push @Aitkens, do {
my @c = \$b[-1];
push @c, \$b[\$_] + \$c[\$_] for 0..\$#b;
@b = @c;
[@c]
} until (@Aitkens == 50);

my @Bell_numbers = map { @\$_[0] } @Aitkens;

say 'First fifteen and fiftieth Bell numbers:';
printf "%2d: %s\n", 1+\$_, \$Bell_numbers[\$_] for 0..14, 49;

say "\nFirst ten rows of Aitken's array:";
printf '%-7d'x@{\$Aitkens[\$_]}."\n", @{\$Aitkens[\$_]} for 0..9;
```
Output:
```First fifteen and fiftieth Bell numbers:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
50: 10726137154573358400342215518590002633917247281

First ten rows of Aitken's array:
1
1      2
2      3      5
5      7      10     15
15     20     27     37     52
52     67     87     114    151    203
203    255    322    409    523    674    877
877    1080   1335   1657   2066   2589   3263   4140
4140   5017   6097   7432   9089   11155  13744  17007  21147
21147  25287  30304  36401  43833  52922  64077  77821  94828  115975
```

## Phix

Library: Phix/mpfr

Started out as a translation of Go, but the main routine has now been completely replaced.

```with javascript_semantics
include mpfr.e
function bellTriangle(integer n)
-- nb: returns strings to simplify output
mpz z = mpz_init(1)
string sz = "1"
sequence tri = {}, line = {}
for i=1 to n do
line = prepend(line,mpz_init_set(z))
tri = append(tri,{sz})
for j=2 to length(line) do
mpz_set(line[j],z)
sz = mpz_get_str(z)
tri[\$] = append(tri[\$],sz)
end for
end for
line = mpz_free(line)
z = mpz_free(z)
return tri
end function

sequence bt = bellTriangle(50)
printf(1,"First fifteen and fiftieth Bell numbers:\n%s\n50:%s\n\n",
{join(vslice(bt[1..15],1)),bt[50][1]})
printf(1,"The first ten rows of Bell's triangle:\n")
for i=1 to 10 do
printf(1,"%s\n",{join(bt[i])})
end for
```
Output:
```First fifteen and fiftieth Bell numbers:
1 1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 190899322
50:10726137154573358400342215518590002633917247281

The first ten rows of Bell's triangle:
1
1 2
2 3 5
5 7 10 15
15 20 27 37 52
52 67 87 114 151 203
203 255 322 409 523 674 877
877 1080 1335 1657 2066 2589 3263 4140
4140 5017 6097 7432 9089 11155 13744 17007 21147
21147 25287 30304 36401 43833 52922 64077 77821 94828 115975
```

## Picat

First 18 Bell numbers and b(50). (Port of the Sage solution at the OEIS A000110 page.)

```main =>
B50=b(50),
println(B50[1..18]),
println(b50=B50.last),
nl.

b(M) = R =>
A = new_array(M-1),
bind_vars(A,0),
A[1] := 1,
R = [1, 1],
foreach(N in 2..M-1)
A[N] := A[1],
foreach(K in N..-1..2)
A[K-1] := A[K-1] + A[K],
end,
R := R ++ [A[1]]
end.```
Output:
```[1,1,2,5,15,52,203,877,4140,21147,115975,678570,4213597,27644437,190899322,1382958545,10480142147,82864869804]
b50 = 10726137154573358400342215518590002633917247281```

Bell's Triangle (and the 50th Bell number)

Translation of: D
```main =>
Tri = tri(50),
foreach(I in 1..10)
println(Tri[I].to_list)
end,
nl,
println(tri50=Tri.last.first),
nl.

% Adjustments for base-1.
tri(N) = Tri[2..N+1] =>
Tri = new_array(N+1),
foreach(I in 1..N+1)
Tri[I] := new_array(I-1),
bind_vars(Tri[I],0)
end,
Tri[2,1] := 1,
foreach(I in 3..N+1)
Tri[I,1] := Tri[I-1,I-2],
foreach(J in 2..I-1)
Tri[I,J] := Tri[I,J-1] + Tri[I-1,J-1]
end
end.```
Output:
```[1]
[1,2]
[2,3,5]
[5,7,10,15]
[15,20,27,37,52]
[52,67,87,114,151,203]
[203,255,322,409,523,674,877]
[877,1080,1335,1657,2066,2589,3263,4140]
[4140,5017,6097,7432,9089,11155,13744,17007,21147]
[21147,25287,30304,36401,43833,52922,64077,77821,94828,115975]

tri50 = 10726137154573358400342215518590002633917247281```
Translation of: Prolog
```main :-
bell(49, Bell),
printf("First 15 Bell numbers:\n"),
print_bell_numbers(Bell, 15),
Number=Bell.last.first,
printf("\n50th Bell number: %w\n", Number),
printf("\nFirst 10 rows of Bell triangle:\n"),
print_bell_rows(Bell, 10).

bell(N, Bell):-
bell(N, Bell, [], _).

bell(0, [[1]|T], T, [1]):-!.
bell(N, Bell, B, Row):-
N1 is N - 1,
bell(N1, Bell, [Row|B], Last),
next_row(Row, Last).

next_row([Last|Bell], Bell1):-
Last=last(Bell1),
next_row1(Last, Bell, Bell1).

next_row1(_, [], []):-!.
next_row1(X, [Y|Rest], [B|Bell]):-
Y is X + B,
next_row1(Y, Rest, Bell).

print_bell_numbers(_, 0):-!.
print_bell_numbers([[Number|_]|Bell], N):-
printf("%w\n", Number),
N1 is N - 1,
print_bell_numbers(Bell, N1).

print_bell_rows(_, 0):-!.
print_bell_rows([Row|Rows], N):-
print_bell_row(Row),
N1 is N - 1,
print_bell_rows(Rows, N1).

print_bell_row([Number]):-
!,
printf("%w\n", Number).
print_bell_row([Number|Numbers]):-
printf("%w ", Number),
print_bell_row(Numbers).```
Output:
```First 15 Bell numbers:
1
1
2
5
15
52
203
877
4140
21147
115975
678570
4213597
27644437
190899322

50th Bell number: 10726137154573358400342215518590002633917247281

First 10 rows of Bell triangle:
1
1 2
2 3 5
5 7 10 15
15 20 27 37 52
52 67 87 114 151 203
203 255 322 409 523 674 877
877 1080 1335 1657 2066 2589 3263 4140
4140 5017 6097 7432 9089 11155 13744 17007 21147
21147 25287 30304 36401 43833 52922 64077 77821 94828 115975```

### Constraint modelling: construct (and count) the sets

This model creates the underlying structure of the sets, and for N=1..4 it also converts and show the proper sets.

What the constraint model really creates is the following for N=3:

```  [[1,1,1],[1,1,2],[1,2,1],[1,2,2],[1,2,3]]
```

where the i'th value in a (sub) list indicates which set it belongs to. E.g. the list

``` [1,2,1]
```

indicates that both 1 and 3 belongs to the first set, and 2 belongs to the second set, i.e.

``` {{1,3},{2}, {}}
```

and after the empty lists are removed:

``` {{1,3},{2}}
```

The full set is converted to

```  {Template:1,2,3,{{1,2},{3}},{{1,3},{2}},{{1},{2,3}},{{1},{2},{3}}}
```

The symmetry constraint `value_precede_chain/2` ensures that a value N+1 is not placed in the list (X) before all the values 1..N has been placed ("seen") in the list. This handles the symmetry that the two sets {1,2} and {2,1} are to be considered the same.

```import cp.

main =>
member(N,1..10),
X = new_list(N),
X :: 1..N,
value_precede_chain(1..N,X),
solve_all(\$[ff,split],X)=All,
println(N=All.len),
if N <= 4 then
% convert to sets
Set = {},
foreach(Y in All)
L = new_array(N),
bind_vars(L,{}),
foreach(I in 1..N)
L[Y[I]] := L[Y[I]] ++ {I}
end,
Set := Set ++ { {E : E in L, E != {}} }
end,
println(Set)
end,
nl,
fail,
nl.

%
% Ensure that a value N+1 is placed in the list X not before
% all the value 1..N are placed in the list.
%
value_precede_chain(C, X) =>
foreach(I in 2..C.length)
value_precede(C[I-1], C[I], X)
end.

value_precede(S,T,X) =>
XLen = X.length,
B = new_list(XLen+1),
B :: 0..1,
foreach(I in 1..XLen)
Xis #= (X[I] #= S),
(Xis #=> (B[I+1] #= 1))
#/\ ((#~ Xis #= 1) #=> (B[I] #= B[I+1]))
#/\ ((#~ B[I] #= 1) #=> (X[I] #!= T))
end,
B[1] #= 0.```
Output:
```1 = 1
{{{1}}}

2 = 2
{{{1,2}},{{1},{2}}}

3 = 5
{{{1,2,3}},{{1,2},{3}},{{1,3},{2}},{{1},{2,3}},{{1},{2},{3}}}

4 = 15
{{{1,2,3,4}},{{1,2,3},{4}},{{1,2,4},{3}},{{1,2},{3,4}},{{1,2},{3},{4}},{{1,3,4},{2}},{{1,3},{2,4}},{{1,3},{2},{4}},{{1,4},{2,3}},{{1},{2,3,4}},{{1},{2,3},{4}},{{1,4},{2},{3}},{{1},{2,4},{3}},{{1},{2},{3,4}},{{1},{2},{3},{4}}}

5 = 52

6 = 203

7 = 877

8 = 4140

9 = 21147

10 = 115975```

## PicoLisp

```(de bell (N)
(make
(setq L (link (list 1)))
(do N
(setq L
(make
(setq A (link (last L)))
(for B L
(setq A (link (+ A B))) ) ) ) ) ) ) )
(setq L (bell 51))
(for N 15
(tab (2 -2 -2) N ":" (get L N 1)) )
(prinl 50 ": " (get L 50 1))
(prinl)
(prinl "First ten rows:")
(for N 10
(println (get L N)) )```
Output:
``` 1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
50: 10726137154573358400342215518590002633917247281

First ten rows:
(1)
(1 2)
(2 3 5)
(5 7 10 15)
(15 20 27 37 52)
(52 67 87 114 151 203)
(203 255 322 409 523 674 877)
(877 1080 1335 1657 2066 2589 3263 4140)
(4140 5017 6097 7432 9089 11155 13744 17007 21147)
(21147 25287 30304 36401 43833 52922 64077 77821 94828 115975)
```

## Prolog

Works with: SWI Prolog
```bell(N, Bell):-
bell(N, Bell, [], _).

bell(0, [[1]|T], T, [1]):-!.
bell(N, Bell, B, Row):-
N1 is N - 1,
bell(N1, Bell, [Row|B], Last),
next_row(Row, Last).

next_row([Last|Bell], Bell1):-
last(Bell1, Last),
next_row1(Last, Bell, Bell1).

next_row1(_, [], []):-!.
next_row1(X, [Y|Rest], [B|Bell]):-
Y is X + B,
next_row1(Y, Rest, Bell).

print_bell_numbers(_, 0):-!.
print_bell_numbers([[Number|_]|Bell], N):-
writef('%w\n', [Number]),
N1 is N - 1,
print_bell_numbers(Bell, N1).

print_bell_rows(_, 0):-!.
print_bell_rows([Row|Rows], N):-
print_bell_row(Row),
N1 is N - 1,
print_bell_rows(Rows, N1).

print_bell_row([Number]):-
!,
writef('%w\n', [Number]).
print_bell_row([Number|Numbers]):-
writef('%w ', [Number]),
print_bell_row(Numbers).

main:-
bell(49, Bell),
writef('First 15 Bell numbers:\n'),
print_bell_numbers(Bell, 15),
last(Bell, [Number|_]),
writef('\n50th Bell number: %w\n', [Number]),
writef('\nFirst 10 rows of Bell triangle:\n'),
print_bell_rows(Bell, 10).
```
Output:
```First 15 Bell numbers:
1
1
2
5
15
52
203
877
4140
21147
115975
678570
4213597
27644437
190899322

50th Bell number: 10726137154573358400342215518590002633917247281

First 10 rows of Bell triangle:
1
1 2
2 3 5
5 7 10 15
15 20 27 37 52
52 67 87 114 151 203
203 255 322 409 523 674 877
877 1080 1335 1657 2066 2589 3263 4140
4140 5017 6097 7432 9089 11155 13744 17007 21147
21147 25287 30304 36401 43833 52922 64077 77821 94828 115975
```

## Python

### Procedural

Translation of: D
Works with: Python version 2.7
```def bellTriangle(n):
tri = [None] * n
for i in xrange(n):
tri[i] = [0] * i
tri[1][0] = 1
for i in xrange(2, n):
tri[i][0] = tri[i - 1][i - 2]
for j in xrange(1, i):
tri[i][j] = tri[i][j - 1] + tri[i - 1][j - 1]
return tri

def main():
bt = bellTriangle(51)
print "First fifteen and fiftieth Bell numbers:"
for i in xrange(1, 16):
print "%2d: %d" % (i, bt[i][0])
print "50:", bt[50][0]
print
print "The first ten rows of Bell's triangle:"
for i in xrange(1, 11):
print bt[i]

main()
```
Output:
```First fifteen and fiftieth Bell numbers:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
50: 10726137154573358400342215518590002633917247281

The first ten rows of Bell's triangle:
[1]
[1, 2]
[2, 3, 5]
[5, 7, 10, 15]
[15, 20, 27, 37, 52]
[52, 67, 87, 114, 151, 203]
[203, 255, 322, 409, 523, 674, 877]
[877, 1080, 1335, 1657, 2066, 2589, 3263, 4140]
[4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147]
[21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975]```

### Functional

Works with: Python version 3.7
```'''Bell numbers'''

from itertools import accumulate, chain, islice
from operator import add, itemgetter
from functools import reduce

# bellNumbers :: [Int]
def bellNumbers():
'''Bell or exponential numbers.
A000110
'''
return map(itemgetter(0), bellTriangle())

# bellTriangle :: [[Int]]
def bellTriangle():
'''Bell triangle.'''
return map(
itemgetter(1),
iterate(
compose(
bimap(last)(identity),
)
)((1, [1]))
)

# ------------------------- TEST --------------------------
# main :: IO ()
def main():
'''Tests'''
showIndex = compose(repr, succ, itemgetter(0))
showValue = compose(repr, itemgetter(1))
print(
fTable(
'First fifteen Bell numbers:'
)(showIndex)(showValue)(identity)(list(
enumerate(take(15)(bellNumbers()))
))
)

print('\nFiftieth Bell number:')
bells = bellNumbers()
drop(49)(bells)
print(
next(bells)
)

print(
fTable(
"\nFirst 10 rows of Bell's triangle:"
)(showIndex)(showValue)(identity)(list(
enumerate(take(10)(bellTriangle()))
))
)

# ------------------------ GENERIC ------------------------

# bimap :: (a -> b) -> (c -> d) -> (a, c) -> (b, d)
def bimap(f):
'''Tuple instance of bimap.
A tuple of the application of f and g to the
first and second values respectively.
'''
def go(g):
def gox(x):
return (f(x), g(x))
return gox
return go

# compose :: ((a -> a), ...) -> (a -> a)
def compose(*fs):
'''Composition, from right to left,
of a series of functions.
'''
def go(f, g):
def fg(x):
return f(g(x))
return fg
return reduce(go, fs, identity)

# drop :: Int -> [a] -> [a]
# drop :: Int -> String -> String
def drop(n):
'''The sublist of xs beginning at
(zero-based) index n.
'''
def go(xs):
if isinstance(xs, (list, tuple, str)):
return xs[n:]
else:
take(n)(xs)
return xs
return go

# fTable :: String -> (a -> String) ->
# (b -> String) -> (a -> b) -> [a] -> String
def fTable(s):
'''Heading -> x display function -> fx display function ->
f -> xs -> tabular string.
'''
def gox(xShow):
def gofx(fxShow):
def gof(f):
def goxs(xs):
ys = [xShow(x) for x in xs]
w = max(map(len, ys))

def arrowed(x, y):
return y.rjust(w, ' ') + ' -> ' + fxShow(f(x))
return s + '\n' + '\n'.join(
map(arrowed, xs, ys)
)
return goxs
return gof
return gofx
return gox

# identity :: a -> a
def identity(x):
'''The identity function.'''
return x

# iterate :: (a -> a) -> a -> Gen [a]
def iterate(f):
'''An infinite list of repeated
applications of f to x.
'''
def go(x):
v = x
while True:
yield v
v = f(v)
return go

# last :: [a] -> a
def last(xs):
'''The last element of a non-empty list.'''
return xs[-1]

# scanl :: (b -> a -> b) -> b -> [a] -> [b]
def scanl(f):
'''scanl is like reduce, but returns a succession of
intermediate values, building from the left.
'''
def go(a):
def g(xs):
return accumulate(chain([a], xs), f)
return g
return go

# succ :: Enum a => a -> a
def succ(x):
'''The successor of a value.
For numeric types, (1 +).
'''
return 1 + x

# take :: Int -> [a] -> [a]
# take :: Int -> String -> String
def take(n):
'''The prefix of xs of length n,
or xs itself if n > length xs.
'''
def go(xs):
return (
xs[0:n]
if isinstance(xs, (list, tuple))
else list(islice(xs, n))
)
return go

# uncurry :: (a -> b -> c) -> ((a, b) -> c)
def uncurry(f):
'''A function over a tuple,
derived from a curried function.
'''
def go(tpl):
return f(tpl[0])(tpl[1])
return go

# MAIN ---
if __name__ == '__main__':
main()
```
Output:
```First fifteen Bell numbers:
1 -> 1
2 -> 1
3 -> 2
4 -> 5
5 -> 15
6 -> 52
7 -> 203
8 -> 877
9 -> 4140
10 -> 21147
11 -> 115975
12 -> 678570
13 -> 4213597
14 -> 27644437
15 -> 190899322

Fiftieth Bell number:
10726137154573358400342215518590002633917247281

First 10 rows of Bell's triangle:
1 -> [1]
2 -> [1, 2]
3 -> [2, 3, 5]
4 -> [5, 7, 10, 15]
5 -> [15, 20, 27, 37, 52]
6 -> [52, 67, 87, 114, 151, 203]
7 -> [203, 255, 322, 409, 523, 674, 877]
8 -> [877, 1080, 1335, 1657, 2066, 2589, 3263, 4140]
9 -> [4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147]
10 -> [21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975]```

## Quackery

```  [ ' [ [ 1 ] ]  ' [ 1 ]
rot 1 - times
[ dup -1 peek nested
swap witheach
[ over -1 peek + join ]
tuck nested join swap ]
drop ]                        is bell's-triangle ( n --> [ )

[ bell's-triangle [] swap
witheach [ 0 peek join ] ]    is bell-numbers    ( n --> [ )

say "First fifteen Bell numbers:" cr
15 bell-numbers echo
cr cr
say "Fiftieth Bell number:" cr
50 bell-numbers -1 peek echo
cr cr
say "First ten rows of Bell's triangle:" cr
10 bell's-triangle witheach [ echo cr ]```
Output:
```First fifteen Bell numbers:
[ 1 1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 190899322 ]

Fiftieth Bell number:
10726137154573358400342215518590002633917247281

First ten rows of Bell's triangle:
[ 1 ]
[ 1 2 ]
[ 2 3 5 ]
[ 5 7 10 15 ]
[ 15 20 27 37 52 ]
[ 52 67 87 114 151 203 ]
[ 203 255 322 409 523 674 877 ]
[ 877 1080 1335 1657 2066 2589 3263 4140 ]
[ 4140 5017 6097 7432 9089 11155 13744 17007 21147 ]
[ 21147 25287 30304 36401 43833 52922 64077 77821 94828 115975 ]
```

## Racket

```#lang racket

(define (build-bell-row previous-row)
(define seed (last previous-row))
(reverse
(let-values (((reversed _) (for/fold ((acc (list seed)) (prev seed))
((pprev previous-row))
(let ((n (+ prev pprev))) (values (cons n acc) n)))))
reversed)))

(define reverse-bell-triangle
(let ((memo (make-hash '((0 . ((1)))))))
(λ (rows) (hash-ref! memo
rows
(λ ()
(let ((prev (reverse-bell-triangle (sub1 rows))))
(cons (build-bell-row (car prev)) prev)))))))

(define bell-triangle (compose reverse reverse-bell-triangle))

(define bell-number (compose caar reverse-bell-triangle))

(module+ main
(map bell-number (range 15))
(bell-number 50)
(bell-triangle 10))
```
Output:
```'(1 1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 190899322)
185724268771078270438257767181908917499221852770
'((1)
(1 2)
(2 3 5)
(5 7 10 15)
(15 20 27 37 52)
(52 67 87 114 151 203)
(203 255 322 409 523 674 877)
(877 1080 1335 1657 2066 2589 3263 4140)
(4140 5017 6097 7432 9089 11155 13744 17007 21147)
(21147 25287 30304 36401 43833 52922 64077 77821 94828 115975)
(115975 137122 162409 192713 229114 272947 325869 389946 467767 562595 678570))
```

## Raku

(formerly Perl 6)

### via Aitken's array

Works with: Rakudo version 2019.03
``` my @Aitkens-array = lazy [1], -> @b {
my @c = @b.tail;
@c.push: @b[\$_] + @c[\$_] for ^@b;
@c
} ... *;

my @Bell-numbers = @Aitkens-array.map: { .head };

say "First fifteen and fiftieth Bell numbers:";
printf "%2d: %s\n", 1+\$_, @Bell-numbers[\$_] for flat ^15, 49;

say "\nFirst ten rows of Aitken's array:";
.say for @Aitkens-array[^10];
```
Output:
```First fifteen and fiftieth Bell numbers:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
50: 10726137154573358400342215518590002633917247281

First ten rows of Aitken's array:
[1]
[1 2]
[2 3 5]
[5 7 10 15]
[15 20 27 37 52]
[52 67 87 114 151 203]
[203 255 322 409 523 674 877]
[877 1080 1335 1657 2066 2589 3263 4140]
[4140 5017 6097 7432 9089 11155 13744 17007 21147]
[21147 25287 30304 36401 43833 52922 64077 77821 94828 115975]```

### via Recurrence relation

Works with: Rakudo version 2019.03
```sub binomial { [*] (\$^n … 0) Z/ 1 .. \$^p }

my @bell = 1, -> *@s { [+] @s »*« @s.keys.map: { binomial(@s-1, \$_) }  } … *;

.say for @bell[^15], @bell[50 - 1];
```
Output:
```(1 1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 190899322)
10726137154573358400342215518590002633917247281```

### via Stirling sums

Works with: Rakudo version 2019.03
```my @Stirling_numbers_of_the_second_kind =
(1,),
{ (0, |@^last) »+« (|(@^last »*« @^last.keys), 0) } … *
;
my @bell = @Stirling_numbers_of_the_second_kind.map: *.sum;

.say for @bell.head(15), @bell[50 - 1];
```
Output:
```(1 1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 190899322)
10726137154573358400342215518590002633917247281 ```

## REXX

Bell numbers are the number of ways of placing   n   labeled balls into   n   indistinguishable boxes.   Bell(0)   is defined as   1.

This REXX version uses an   index   of the Bell number   (which starts a zero).

A little optimization was added in calculating the factorial of a number by using memoization.

Also, see this task's   discussion   to view how the sizes of Bell numbers increase in relation to its index.

```/*REXX program calculates and displays a range of  Bell numbers  (index starts at zero).*/
parse arg LO HI .                                /*obtain optional arguments from the CL*/
if LO=='' & HI==""   then do; LO=0; HI=14;  end  /*Not specified?  Then use the default.*/
if LO=='' | LO==","  then LO=  0                 /* "      "         "   "   "     "    */
if HI=='' | HI==","  then HI= 15                 /* "      "         "   "   "     "    */
numeric digits max(9, HI*2)                      /*crudely calculate the # decimal digs.*/
!.=.;             !.0= 1;   !.1= 1;      @.= 1   /*the  FACT  function uses memoization.*/
do j=0  for  HI+1;     \$= j==0;     jm= j-1 /*JM  is used for a shortcut  (below). */
do k=0  for j;            _= jm - k  /* [↓]  calculate a Bell # the easy way*/
\$= \$ + comb(jm, k) * @._             /*COMB≡combination or binomial function*/
end   /*k*/
@.j= \$                                      /*assign the Jth Bell number to @ array*/
if j>=LO  &  j<=HI  then say '      Bell('right(j, length(HI) )") = "    commas(\$)
end   /*j*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas:  parse arg _;  do c=length(_)-3  to 1  by -3; _=insert(',', _, c); end;   return _
/*──────────────────────────────────────────────────────────────────────────────────────*/
comb: procedure expose !.; parse arg x,y;        if x==y      then  return 1
if x-y<y  then y= x - y;                   if !.x.y\==. then  return !.x.y / fact(y)
_= 1;          do j=x-y+1  to x;  _= _*j;  end;    !.x.y= _;  return     _ / fact(y)
/*──────────────────────────────────────────────────────────────────────────────────────*/
fact: procedure expose !.; parse arg x;    if !.x\==.   then return !.x;          != 1
do f=2  for x-1;      != ! * f;    end;        !.x= !;       return !
```
output   when using the internal default inputs of:     0   14
```      Bell( 0) =  1
Bell( 1) =  1
Bell( 2) =  2
Bell( 3) =  5
Bell( 4) =  15
Bell( 5) =  52
Bell( 6) =  203
Bell( 7) =  877
Bell( 8) =  4,140
Bell( 9) =  21,147
Bell(10) =  115,975
Bell(11) =  678,570
Bell(12) =  4,213,597
Bell(13) =  27,644,437
Bell(14) =  190,899,322
```
output   when using the inputs of:     49   49
```      Bell(49) =  10,726,137,154,573,358,400,342,215,518,590,002,633,917,247,281
```

## RPL

Works with: HP version 48
```≪ → n
≪ { 1 }
WHILE 'n' DECR REPEAT
DUP DUP SIZE GET 1 →LIST
1 3 PICK SIZE FOR j
OVER j GET OVER j GET + +
NEXT SWAP DROP
≫ ≫ ‘BELL’ STO
```

Variant with a better use of the stack

Slightly faster then, although more wordy:

```≪ → n
≪ { 1 } 1
WHILE 'n' DECR REPEAT
DUP 1 →LIST
1 4 PICK SIZE FOR j
3 PICK j GET ROT + SWAP OVER +
NEXT ROT DROP SWAP
≫ ≫ ‘BELL’ STO
```
```≪ {} 1 15 FOR n n BELL + NEXT ≫ EVAL
```
Output:
```1: { 1 1 2 5 15 52 203 877 4140 21147 115975 678570 4213597 27644437 190899322 }
```

It makes no sense to display 10 rows of the Bell triangle on a screen limited to 22 characters and 7 lines in the best case.

## Ruby

Translation of: D
```def bellTriangle(n)
tri = Array.new(n)
for i in 0 .. n - 1 do
tri[i] = Array.new(i)
for j in 0 .. i - 1 do
tri[i][j] = 0
end
end
tri[1][0] = 1
for i in 2 .. n - 1 do
tri[i][0] = tri[i - 1][i - 2]
for j in 1 .. i - 1 do
tri[i][j] = tri[i][j - 1] + tri[i - 1][j - 1]
end
end
return tri
end

def main
bt = bellTriangle(51)
puts "First fifteen and fiftieth Bell numbers:"
for i in 1 .. 15 do
puts "%2d: %d" % [i, bt[i][0]]
end
puts "50: %d" % [bt[50][0]]
puts

puts "The first ten rows of Bell's triangle:"
for i in 1 .. 10 do
puts bt[i].inspect
end
end

main()
```
Output:
```First fifteen and fiftieth Bell numbers:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
50: 10726137154573358400342215518590002633917247281

The first ten rows of Bell's triangle:
[1]
[1, 2]
[2, 3, 5]
[5, 7, 10, 15]
[15, 20, 27, 37, 52]
[52, 67, 87, 114, 151, 203]
[203, 255, 322, 409, 523, 674, 877]
[877, 1080, 1335, 1657, 2066, 2589, 3263, 4140]
[4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147]
[21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975]```

## Rust

Translation of: D
Library: num version 0.2
```use num::BigUint;

fn main() {
let bt = bell_triangle(51);
// the fifteen first
for i in 1..=15 {
println!("{}: {}", i, bt[i][0]);
}

// the fiftieth
println!("50: {}", bt[50][0])
}

fn bell_triangle(n: usize) -> Vec<Vec<BigUint>> {
let mut tri: Vec<Vec<BigUint>> = Vec::with_capacity(n);
for i in 0..n {
let v = vec![BigUint::from(0u32); i];
tri.push(v);
}
tri[1][0] = BigUint::from(1u32);

for i in 2..n {
tri[i][0] = BigUint::from_bytes_be(&tri[i - 1][i - 2].to_bytes_be());
for j in 1..i {
let added_big_uint = &tri[i][j - 1] + &tri[i - 1][j - 1];
}
}

tri
}
```
Output:
```1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
50: 10726137154573358400342215518590002633917247281
```

## Scala

Translation of: Java
```import scala.collection.mutable.ListBuffer

object BellNumbers {
class BellTriangle {
val arr: ListBuffer[Int] = ListBuffer.empty[Int]

def this(n: Int) {
this()

val length = n * (n + 1) / 2
for (_ <- 0 until length) {
arr += 0
}

this (1, 0) = 1
for (i <- 2 to n) {
this (i, 0) = this (i - 1, i - 2)
for (j <- 1 until i) {
this (i, j) = this (i, j - 1) + this (i - 1, j - 1)
}
}
}

private def index(row: Int, col: Int): Int = {
require(row > 0, "row must be greater than zero")
require(col >= 0, "col must not be negative")
require(col < row, "col must be less than row")

row * (row - 1) / 2 + col
}

def apply(row: Int, col: Int): Int = {
val i = index(row, col)
arr(i)
}

def update(row: Int, col: Int, value: Int): Unit = {
val i = index(row, col)
arr(i) = value
}
}

def main(args: Array[String]): Unit = {
val rows = 15
val bt = new BellTriangle(rows)

println("First fifteen Bell numbers:")
for (i <- 0 until rows) {
printf("%2d: %d\n", i + 1, bt(i + 1, 0))
}

for (i <- 1 to 10) {
print(bt(i, 0))
for (j <- 1 until i) {
print(s", \${bt(i, j)}")
}
println()
}
}
}
```
Output:
```First fifteen Bell numbers:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
1
1, 2
2, 3, 5
5, 7, 10, 15
15, 20, 27, 37, 52
52, 67, 87, 114, 151, 203
203, 255, 322, 409, 523, 674, 877
877, 1080, 1335, 1657, 2066, 2589, 3263, 4140
4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147
21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975```

## Scheme

Works with: Chez Scheme
```; Given the remainder of the previous row and the final cons of the current row,
; extend (in situ) the current row to be a complete row of the Bell triangle.
; Return the final value in the extended row (for use in computing the following row).

(define bell-triangle-row-extend
(lambda (prevrest thisend)
(cond
((null? prevrest)
(car thisend))
(else
(set-cdr! thisend (list (+ (car prevrest) (car thisend))))
(bell-triangle-row-extend (cdr prevrest) (cdr thisend))))))

; Return the Bell triangle of rows 0 through N (as a list of lists).

(define bell-triangle
(lambda (n)
(let* ((tri (list (list 1)))
(triend tri)
(rowendval (caar tri)))
(do ((index 0 (1+ index)))
((>= index n) tri)
(let ((nextrow (list rowendval)))
(set! rowendval (bell-triangle-row-extend (car triend) nextrow))
(set-cdr! triend (list nextrow))
(set! triend (cdr triend)))))))

; Print out the Bell numbers 0 through 19 and 49 thgough 51.
; (The Bell numbers are the first number on each row of the Bell triangle.)

(printf "~%The Bell numbers:~%")
(let loop ((triangle (bell-triangle 51)) (count 0))
(when (pair? triangle)
(when (or (<= count 19) (>= count 49))
(printf "  ~2d: ~:d~%" count (caar triangle)))
(loop (cdr triangle) (1+ count))))

; Print out the Bell triangle of 10 rows.

(printf "~%First 10 rows of the Bell triangle:~%")
(let rowloop ((triangle (bell-triangle 9)))
(when (pair? triangle)
(let eleloop ((rowlist (car triangle)))
(when (pair? rowlist)
(printf " ~7:d" (car rowlist))
(eleloop (cdr rowlist))))
(newline)
(rowloop (cdr triangle))))
```
Output:
```The Bell numbers:
0: 1
1: 1
2: 2
3: 5
4: 15
5: 52
6: 203
7: 877
8: 4,140
9: 21,147
10: 115,975
11: 678,570
12: 4,213,597
13: 27,644,437
14: 190,899,322
15: 1,382,958,545
16: 10,480,142,147
17: 82,864,869,804
18: 682,076,806,159
19: 5,832,742,205,057
49: 10,726,137,154,573,358,400,342,215,518,590,002,633,917,247,281
50: 185,724,268,771,078,270,438,257,767,181,908,917,499,221,852,770
51: 3,263,983,870,004,111,524,856,951,830,191,582,524,419,255,819,477

First 10 rows of the Bell triangle:
1
1       2
2       3       5
5       7      10      15
15      20      27      37      52
52      67      87     114     151     203
203     255     322     409     523     674     877
877   1,080   1,335   1,657   2,066   2,589   3,263   4,140
4,140   5,017   6,097   7,432   9,089  11,155  13,744  17,007  21,147
21,147  25,287  30,304  36,401  43,833  52,922  64,077  77,821  94,828 115,975```

## Sidef

Built-in:

```say 15.of { .bell }
```

Formula as a sum of Stirling numbers of the second kind:

```func bell(n) { sum(0..n, {|k| stirling2(n, k) }) }
```

Via Aitken's array (optimized for space):

```func bell_numbers (n) {

var acc = []
var bell = [1]

(n-1).times {
acc.unshift(bell[-1])
acc.accumulate!
bell.push(acc[-1])
}

bell
}

var B = bell_numbers(50)
say "The first 15 Bell numbers: #{B.first(15).join(', ')}"
say "The fiftieth Bell number : #{B[50-1]}"
```
Output:
```The first 15 Bell numbers: 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322
The fiftieth Bell number : 10726137154573358400342215518590002633917247281
```

Aitken's array:

```func aitken_array (n) {

var A = [1]

[[1]] + (n-1).of {
A = [A[-1], A...].accumulate
}
}

aitken_array(10).each { .say }
```
Output:
```[1]
[1, 2]
[2, 3, 5]
[5, 7, 10, 15]
[15, 20, 27, 37, 52]
[52, 67, 87, 114, 151, 203]
[203, 255, 322, 409, 523, 674, 877]
[877, 1080, 1335, 1657, 2066, 2589, 3263, 4140]
[4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147]
[21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975]
```

Aitken's array (recursive definition):

```func A((0), (0))       { 1 }
func A(n, (0))         { A(n-1, n-1) }
func A(n, k) is cached { A(n, k-1) + A(n-1, k-1) }

for n in (^10) {
say (0..n -> map{|k| A(n, k) })
}
```

(same output as above)

## Swift

Translation of: Kotlin
```public struct BellTriangle<T: BinaryInteger> {
@usableFromInline
var arr: [T]

@inlinable
public internal(set) subscript(row row: Int, col col: Int) -> T {
get { arr[row * (row - 1) / 2 + col] }
set { arr[row * (row - 1) / 2 + col] = newValue }
}

@inlinable
public init(n: Int) {
arr = Array(repeating: 0, count: n * (n + 1) / 2)

self[row: 1, col: 0] = 1

for i in 2...n {
self[row: i, col: 0] = self[row: i - 1, col: i - 2]

for j in 1..<i {
self[row: i, col: j] = self[row: i, col: j - 1] + self[row: i - 1, col: j - 1]
}
}
}
}

let tri = BellTriangle<Int>(n: 15)

print("First 15 Bell numbers:")

for i in 1...15 {
print("\(i): \(tri[row: i, col: 0])")
}

for i in 1...10 {
print(tri[row: i, col: 0], terminator: "")

for j in 1..<i {
print(", \(tri[row: i, col: j])", terminator: "")
}

print()
}
```
Output:
```First 15 Bell numbers:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
1
1, 2
2, 3, 5
5, 7, 10, 15
15, 20, 27, 37, 52
52, 67, 87, 114, 151, 203
203, 255, 322, 409, 523, 674, 877
877, 1080, 1335, 1657, 2066, 2589, 3263, 4140
4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147
21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975```

## V (Vlang)

Translation of: Go
```import math.big

fn bell_triangle(n int) [][]big.Integer {
mut tri := [][]big.Integer{len: n}
for i in 0..n {
tri[i] = []big.Integer{len: i}
for j in 0..i {
tri[i][j] = big.zero_int
}
}
tri[1][0] = big.integer_from_u64(1)
for i in 2..n {
tri[i][0] = tri[i-1][i-2]
for j := 1; j < i; j++ {
tri[i][j] = tri[i][j-1] + tri[i-1][j-1]
}
}
return tri
}

fn main() {
bt := bell_triangle(51)
println("First fifteen and fiftieth Bell numbers:")
for i := 1; i <= 15; i++ {
println("\${i:2}: \${bt[i][0]}")
}
println("50: \${bt[50][0]}")
println("\nThe first ten rows of Bell's triangle:")
for i := 1; i <= 10; i++ {
println(bt[i])
}
}```
Output:
```First fifteen and fiftieth Bell numbers:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4140
10: 21147
11: 115975
12: 678570
13: 4213597
14: 27644437
15: 190899322
50: 10726137154573358400342215518590002633917247281

The first ten rows of Bell's triangle:
[1]
[1, 2]
[2, 3, 5]
[5, 7, 10, 15]
[15, 20, 27, 37, 52]
[52, 67, 87, 114, 151, 203]
[203, 255, 322, 409, 523, 674, 877]
[877, 1080, 1335, 1657, 2066, 2589, 3263, 4140]
[4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147]
[21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975
```

## Wren

Translation of: Go
Library: Wren-fmt
```import "./big" for BigInt
import "./fmt" for Fmt

var bellTriangle = Fn.new { |n|
var tri = List.filled(n, null)
for (i in 0...n) {
tri[i] = List.filled(i, null)
for (j in 0...i) tri[i][j] = BigInt.zero
}
tri[1][0] = BigInt.one
for (i in 2...n) {
tri[i][0] = tri[i-1][i-2]
for (j in 1...i) {
tri[i][j] = tri[i][j-1] + tri[i-1][j-1]
}
}
return tri
}

var bt = bellTriangle.call(51)
System.print("First fifteen and fiftieth Bell numbers:")
for (i in 1..15) Fmt.print("\$2d: \$,i", i, bt[i][0])
Fmt.print("\$2d: \$,i", 50, bt[50][0])
System.print("\nThe first ten rows of Bell's triangle:")
for (i in 1..10) Fmt.print("\$,7i", bt[i])
```
Output:
```First fifteen and fiftieth Bell numbers:
1: 1
2: 1
3: 2
4: 5
5: 15
6: 52
7: 203
8: 877
9: 4,140
10: 21,147
11: 115,975
12: 678,570
13: 4,213,597
14: 27,644,437
15: 190,899,322
50: 10,726,137,154,573,358,400,342,215,518,590,002,633,917,247,281

The first ten rows of Bell's triangle:
1
1       2
2       3       5
5       7      10      15
15      20      27      37      52
52      67      87     114     151     203
203     255     322     409     523     674     877
877   1,080   1,335   1,657   2,066   2,589   3,263   4,140
4,140   5,017   6,097   7,432   9,089  11,155  13,744  17,007  21,147
21,147  25,287  30,304  36,401  43,833  52,922  64,077  77,821  94,828 115,975
```

## XPL0

Translation of: Delphi

32-bit integer are required to calculate the first 15 Bell numbers.

Works with: EXPL-32
```\Bell numbers
code CrLf=9, IntOut=11, Text=12;
define MaxN = 14;
integer A(MaxN), I, J, N;

begin
for I:= 0 to MaxN - 1 do A(I):= 0;
N:= 0; A(0):= 1;
Text(0, "B("); IntOut(0, N); Text(0, ") = "); IntOut(0, A(0)); CrLf(0);
while N < MaxN do
begin
A(N):= A(0);
for J:= N downto 1 do A(J - 1):= A(J - 1) + A(J);
N:= N + 1;
Text(0, "B("); IntOut(0, N); Text(0, ") = "); IntOut(0, A(0)); CrLf(0)
end;
end```
Output:
```B(0) = 1
B(1) = 1
B(2) = 2
B(3) = 5
B(4) = 15
B(5) = 52
B(6) = 203
B(7) = 877
B(8) = 4140
B(9) = 21147
B(10) = 115975
B(11) = 678570
B(12) = 4213597
B(13) = 27644437
B(14) = 190899322
```

## zkl

```fcn bellTriangleW(start=1,wantRow=False){	// --> iterator
Walker.zero().tweak('wrap(row){
row.insert(0,row[-1]);
foreach i in ([1..row.len()-1]){ row[i]+=row[i-1] }
wantRow and row or row[-1]
}.fp(List(start))).push(start,start);
}```
```println("First fifteen Bell numbers:");
bellTriangleW().walk(15).println();```
Output:
```First fifteen Bell numbers:
L(1,1,2,5,15,52,203,877,4140,21147,115975,678570,4213597,27644437,190899322)
```
```println("Rows of the Bell Triangle:");
bt:=bellTriangleW(1,True); do(11){ println(bt.next()) }```
Output:
```Rows of the Bell Triangle:
1
1
L(1,2)
L(2,3,5)
L(5,7,10,15)
L(15,20,27,37,52)
L(52,67,87,114,151,203)
L(203,255,322,409,523,674,877)
L(877,1080,1335,1657,2066,2589,3263,4140)
L(4140,5017,6097,7432,9089,11155,13744,17007,21147)
L(21147,25287,30304,36401,43833,52922,64077,77821,94828,115975)
```
Library: GMP
GNU Multiple Precision Arithmetic Library
```print("The fiftieth Bell number: ");
var [const] BI=Import("zklBigNum");  // libGMP
bellTriangleW(BI(1)).drop(50).value.println();```
Output:
```The fiftieth Bell number: 10726137154573358400342215518590002633917247281
```