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A307524 Expansion of e.g.f. Product_{k>=1} (1 + log(1/(1 - x))^k)/(1 - log(1/(1 - x))^k).

%I A307524 #10 Apr 13 2019 04:26:10
%S A307524 1,2,10,76,724,8368,113792,1771824,31001424,601677888,12818974848,
%T A307524 297223165248,7446226027584,200354793323904,5760239869401984,
%U A307524 176170480317568512,5709535272618925824,195419487662892221184,7042458625343222876928,266500916470984705887744
%N A307524 Expansion of e.g.f. Product_{k>=1} (1 + log(1/(1 - x))^k)/(1 - log(1/(1 - x))^k).
%C A307524 Exponential convolution of A320349 and A320350.
%F A307524 E.g.f.: exp(Sum_{k>=1} (sigma(2*k) - sigma(k))*log(1/(1 - x))^k/k).
%F A307524 E.g.f.: 1/theta_4(log(1/(1 - x))).
%F A307524 a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A015128(k)*k!.
%F A307524 a(n) ~ sqrt(Pi) * exp(Pi*sqrt(n/(exp(1)-1)) + Pi^2/(8*(exp(1)-1))) * n^(n - 1/2) / (2^(5/2) * (exp(1)-1)^n). - _Vaclav Kotesovec_, Apr 13 2019
%t A307524 nmax = 19; CoefficientList[Series[Product[(1 + Log[1/(1 - x)]^k)/(1 - Log[1/(1 - x)]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t A307524 nmax = 19; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2 k] - DivisorSigma[1, k]) Log[1/(1 - x)]^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t A307524 nmax = 19; CoefficientList[Series[1/EllipticTheta[4, 0, Log[1/(1 - x)]], {x, 0, nmax}], x] Range[0, nmax]!
%t A307524 Table[Sum[Abs[StirlingS1[n, k]] Sum[PartitionsQ[j] PartitionsP[k - j], {j, 0, k}] k!, {k, 0, n}], {n, 0, 19}]
%Y A307524 Cf. A015128, A306045, A307523, A320349, A320350.
%K A307524 nonn
%O A307524 0,2
%A A307524 _Ilya Gutkovskiy_, Apr 12 2019