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A356494 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 A356494 #11 Aug 09 2022 11:20:15
%S A356494 1,1,6,35,327,2892,37943,459895,7330172,116054835,2168292295,
%T A356494 41072348550,898738816957,19782331776937,487091519709590,
%U A356494 12305361661242275,337777113607935587,9528258228302443724,289373132780801591323,9016757353084706862647
%N A356494 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 A356494 a(0) = 1; a(n) = Sum_{k=1..n} A354843(k) * binomial(n-1,k-1) * a(n-k).
%o A356494 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(k*x^k)-1))))
%o A356494 (PARI) a354843(n) = n!*sumdiv(n, d, (n/d)^d/d!);
%o A356494 a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354843(j)*binomial(i-1, j-1)*v[i-j+1])); v;
%Y A356494 Cf. A209903, A346055, A356495.
%Y A356494 Cf. A000110, A354843, A356460.
%K A356494 nonn
%O A356494 0,3
%A A356494 _Seiichi Manyama_, Aug 09 2022