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A356495 Expansion of e.g.f. Product_{k>0} B((k * x)^k) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

%I A356495 #8 Aug 09 2022 11:20:47
%S A356495 1,1,10,191,7287,424292,37434683,4512452023,726390985036,
%T A356495 149098938941283,38187088904721655,11903871288193251930,
%U A356495 4442392007373264794677,1953788894138983864638457,1000334575509506861927067378,589712001176601700420819946615
%N A356495 Expansion of e.g.f. Product_{k>0} B((k * x)^k) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.
%F A356495 a(0) = 1; a(n) = Sum_{k=1..n} A354892(k) * binomial(n-1,k-1) * a(n-k).
%o A356495 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp((k*x)^k)-1))))
%o A356495 (PARI) a354892(n) = n!*sumdiv(n, d, d^n/(n/d)!);
%o A356495 a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354892(j)*binomial(i-1, j-1)*v[i-j+1])); v;
%Y A356495 Cf. A209903, A346055, A356494.
%Y A356495 Cf. A000110, A354892.
%K A356495 nonn
%O A356495 0,3
%A A356495 _Seiichi Manyama_, Aug 09 2022