cp's OEIS Frontend

A264432 Third-order Bell numbers.

%I A264432 #30 Jan 30 2021 01:50:10
%S A264432 1,1,2,6,24,119,700,4748,36403,310851,2922606,29977587,332929492,
%T A264432 3978258079,50872884285,692985674373,10015172966221,153021613683924,
%U A264432 2464031776132958,41698912656882644,739771703127828419,13727160292457369098,265876635231121617716
%N A264432 Third-order Bell numbers.
%H A264432 Alois P. Heinz, <a href="/A264432/b264432.txt">Table of n, a(n) for n = 0..469</a>
%H A264432 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>
%p A264432 b:= proc(n, h) option remember; `if`(min(n, h)=0, 1, add(
%p A264432       binomial(n-1, j-1)*b(j-1, h-1)*b(n-j, h), j=1..n))
%p A264432     end:
%p A264432 a:= n-> b(n, 3):
%p A264432 seq(a(n), n=0..22);  # _Alois P. Heinz_, Aug 21 2017
%t A264432 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 *)
%o A264432 (Sage) # uses[bell_transform from A264428]
%o A264432 def A264432_list(dim):
%o A264432     uno = [1]*dim
%o A264432     bell_number = [sum(bell_transform(n, uno)) for n in range(dim)]
%o A264432     bell_number_2 = [sum(bell_transform(n, bell_number)) for n in range(dim)]
%o A264432     return [sum(bell_transform(n, bell_number_2)) for n in range(dim)]
%o A264432 print(A264432_list(23))
%o A264432 (PARI)
%o A264432 \\ For n>23 precision has to be adapted as needed!
%o A264432 A = exp('x + O('x^33) );
%o A264432 B = exp( intformal(A) );
%o A264432 C = exp( intformal(B) );
%o A264432 D = exp( intformal(C) );
%o A264432 Vec( serlaplace(D) )
%o A264432 (Python)
%o A264432 from sympy.core.cache import cacheit
%o A264432 from sympy import binomial
%o A264432 @cacheit
%o A264432 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))
%o A264432 def a(n): return b(n, 3)
%o A264432 print([a(n) for n in range(31)]) # _Indranil Ghosh_, Aug 21 2017, after Maple code
%Y A264432 Cf. A000110, A187761, A264428, A265312.
%K A264432 nonn
%O A264432 0,3
%A A264432 _Peter Luschny_, Dec 02 2015