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A356461 Expansion of e.g.f. ( Product_{k>0} B(x^k)^k )^(1/(1-x)) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

%I A356461 #10 Aug 08 2022 09:40:03
%S A356461 1,1,8,62,631,7417,104489,1648845,29319588,572982162,12250559615,
%T A356461 283321630605,7053444523393,187711377451249,5317981377046420,
%U A356461 159652557864884330,5061465465801168419,168886786171198864725,5914650120884760212977,216844308186908733542877
%N A356461 Expansion of e.g.f. ( Product_{k>0} B(x^k)^k )^(1/(1-x)) where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.
%F A356461 a(0) = 1; a(n) = Sum_{k=1..n} A356459(k) * binomial(n-1,k-1) * a(n-k).
%o A356461 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(x^k)-1)^k)^(1/(1-x))))
%o A356461 (PARI) a356459(n) = n!*sum(k=1, n, sumdiv(k, d, d/(k/d)!));
%o A356461 a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356459(j)*binomial(i-1, j-1)*v[i-j+1])); v;
%Y A356461 Cf. A356025, A356458.
%Y A356461 Cf. A000110, A356459, A356460.
%K A356461 nonn
%O A356461 0,3
%A A356461 _Seiichi Manyama_, Aug 08 2022