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