A264432 Third-order Bell numbers.

1, 1, 2, 6, 24, 119, 700, 4748, 36403, 310851, 2922606, 29977587, 332929492, 3978258079, 50872884285, 692985674373, 10015172966221, 153021613683924, 2464031776132958, 41698912656882644, 739771703127828419, 13727160292457369098, 265876635231121617716

Offset: 0

Author

Peter Luschny, Dec 02 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..469
• Peter Luschny, The Bell transform

Crossrefs

Cf. A000110, A187761, A264428, A265312.

Programs

Maple

b:= proc(n, h) option remember; `if`(min(n, h)=0, 1, add(
      binomial(n-1, j-1)*b(j-1, h-1)*b(n-j, h), j=1..n))
    end:
a:= n-> b(n, 3):
seq(a(n), n=0..22);  # Alois P. Heinz, Aug 21 2017

Mathematica

b[n_, h_]:=b[n, h]=If[Min[n, h]==0, 1, Sum[Binomial[n - 1, j - 1] b[j - 1, h - 1] b[n - j, h] , {j, n}]]; Table[b[n, 3], {n, 0, 30}] (* Indranil Ghosh, Aug 21 2017, after Maple code *)

PARI

\\ For n>23 precision has to be adapted as needed!
A = exp('x + O('x^33) );
B = exp( intformal(A) );
C = exp( intformal(B) );
D = exp( intformal(C) );
Vec( serlaplace(D) )

Python

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def b(n, h): return 1 if min(n, h)==0 else sum(binomial(n - 1, j - 1)*b(j - 1, h - 1)*b(n - j, h) for j in range(1, n + 1))
def a(n): return b(n, 3)
print([a(n) for n in range(31)]) # Indranil Ghosh, Aug 21 2017, after Maple code

Sage

# uses[bell_transform from A264428]
def A264432_list(dim):
    uno = [1]*dim
    bell_number = [sum(bell_transform(n, uno)) for n in range(dim)]
    bell_number_2 = [sum(bell_transform(n, bell_number)) for n in range(dim)]
    return [sum(bell_transform(n, bell_number_2)) for n in range(dim)]
print(A264432_list(23))