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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..469

Peter Luschny, The Bell transform

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 *)

Prog

(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))
(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

Crossrefs

Cf. A000110, A187761, A264428, A265312.

Sequence in context: A047889 A256207 A256208 * A094198 A297200 A071077

Adjacent sequences: A264429 A264430 A264431 * A264433 A264434 A264435

Keyword

nonn

Author

Peter Luschny, Dec 02 2015

Status

approved