%I #16 Jan 23 2019 02:39:42

%S 1,1,5,32,278,2894,35986,514128,8306448,149558688,2968216944,

%T 64314676128,1510065781968,38178537908016,1033794746169168,

%U 29840453678758272,914461132860063360,29645845798652997120,1013511411165693991680,36436289007997132646400,1373976152501162688288000

%N Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1/(1 - x))^k).

%H Robert Israel, <a href="/A320349/b320349.txt">Table of n, a(n) for n = 0..413</a>

%F E.g.f.: exp(Sum_{k>=1} sigma(k)*log(1/(1 - x))^k/k).

%F a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A000041(k)*k!.

%F From _Vaclav Kotesovec_, Oct 13 2018: (Start)

%F a(n) ~ n! * exp(n + Pi*sqrt(2*n/(3*(exp(1) - 1))) + Pi^2/(12*(exp(1) - 1))) / (4 * sqrt(3) * n * (exp(1) - 1)^n).

%F a(n) ~ sqrt(Pi) * exp(Pi*sqrt(2*n/(3*(exp(1) - 1))) + Pi^2/(12*(exp(1) - 1))) * n^(n - 1/2) / (2^(3/2) * sqrt(3) * (exp(1) - 1)^n).

%F (End)

%p seq(n!*coeff(series(mul(1/(1-log(1/(1-x))^k),k=1..100),x=0,21),x,n),n=0..20); # _Paolo P. Lava_, Jan 09 2019

%t nmax = 20; CoefficientList[Series[Product[1/(1 - Log[1/(1 - x)]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

%t nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] Log[1/(1 - x)]^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[Abs[StirlingS1[n, k]] PartitionsP[k] k!, {k, 0, n}], {n, 0, 20}]

%Y Cf. A000041, A000203, A048994, A053529, A167137, A306042, A320350.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Oct 11 2018