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%I #9 Jan 09 2019 09:14:08
%S 1,1,3,20,148,1384,15728,207696,3094152,51423288,945943512,
%T 19083180192,418550811600,9907493349168,251588827187280,
%U 6820899616891008,196645361557479552,6007407711127690752,193842462200078260224,6586904673329133618432,235079477736802622742528,8790132360155070084076800
%N Expansion of e.g.f. Product_{k>=1} (1 + log(1/(1 - x))^k).
%F a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A000009(k)*k!.
%F From _Vaclav Kotesovec_, Oct 13 2018: (Start)
%F a(n) ~ n! * exp(n + Pi*sqrt(n/(3*(exp(1) - 1))) + Pi^2/(24*(exp(1) - 1))) / (4 * 3^(1/4) * n^(3/4) * (exp(1) - 1)^(n + 1/4)).
%F a(n) ~ sqrt(Pi) * exp(Pi*sqrt(n/(3*(exp(1) - 1))) + Pi^2/(24*(exp(1) - 1))) * n^(n - 1/4) / (2^(3/2) * 3^(1/4) * (exp(1) - 1)^(n + 1/4)).
%F (End)
%p seq(n!*coeff(series(mul((1 + log(1/(1 - x))^k),k=1..100),x=0,22),x,n),n=0..21); # _Paolo P. Lava_, Jan 09 2019
%t nmax = 21; CoefficientList[Series[Product[(1 + Log[1/(1 - x)]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[Abs[StirlingS1[n, k]] PartitionsQ[k] k!, {k, 0, n}], {n, 0, 21}]
%Y Cf. A000009, A048994, A088311, A298905, A305550, A320349.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Oct 11 2018
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