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A306046 Expansion of e.g.f. Product_{k>=1} 1/(1 - (exp(x) - 1)^k)^k.

%I A306046 #14 Mar 08 2024 12:00:31
%S A306046 1,1,7,55,571,6991,101467,1682815,31370731,648823951,14728727227,
%T A306046 363609116575,9692252794891,277304683729711,8471938268282587,
%U A306046 275137855204310335,9461893931226763051,343394421233354232271,13112532730352768439547,525396814643685317840095
%N A306046 Expansion of e.g.f. Product_{k>=1} 1/(1 - (exp(x) - 1)^k)^k.
%H A306046 Vaclav Kotesovec, <a href="/A306046/b306046.txt">Table of n, a(n) for n = 0..400</a>
%H A306046 Vaclav Kotesovec, <a href="/A306046/a306046.jpg">Graph - The asymptotic ratio (20000 terms)</a>
%F A306046 a(n) = Sum_{k=0..n} Stirling2(n,k) * A000219(k) * k!.
%F A306046 a(n) ~ n! * exp(3 * Zeta(3)^(1/3) * n^(2/3) / (2^(4/3) * log(2)^(2/3)) + (1 - log(2)) * Zeta(3)^(2/3) * n^(1/3) / (2^(5/3) * log(2)^(4/3)) - (log(2)^2 + log(2) - 1) * Zeta(3) / (12 * log(2)^2) + 1/12) * Zeta(3)^(7/36) / (A * 2^(11/18) * sqrt(3*Pi) * n^(25/36) * (log(2))^(n + 11/36)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Jun 22 2018
%t A306046 nmax = 20; CoefficientList[Series[Product[1/(1 - (Exp[x] - 1)^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
%Y A306046 Cf. A000219, A167137, A305550, A306045, A306081.
%K A306046 nonn
%O A306046 0,3
%A A306046 _Vaclav Kotesovec_, Jun 18 2018