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A356458 Expansion of e.g.f. ( Product_{k>0} B(x^k) )^(1/(1-x)) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

%I A356458 #13 Aug 08 2022 09:39:49
%S A356458 1,1,6,38,319,3117,36359,476121,7025708,114118746,2029450055,
%T A356458 39078892305,810834093733,17998186069489,425672049713174,
%U A356458 10676653292086790,283014906314277059,7901659174554937925,231719030698518379003,7118469816302381503209
%N A356458 Expansion of e.g.f. ( Product_{k>0} B(x^k) )^(1/(1-x)) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.
%F A356458 a(0) = 1; a(n) = Sum_{k=1..n} A355886(k) * binomial(n-1,k-1) * a(n-k).
%o A356458 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(x^k)-1))^(1/(1-x))))
%o A356458 (PARI) a355886(n) = n!*sum(k=1, n, n\k/k!);
%o A356458 a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a355886(j)*binomial(i-1, j-1)*v[i-j+1])); v;
%Y A356458 Cf. A356025, A356461.
%Y A356458 Cf. A000110, A209903, A355886.
%K A356458 nonn
%O A356458 0,3
%A A356458 _Seiichi Manyama_, Aug 08 2022