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A306081 Expansion of e.g.f. Product_{k>=1} ((1 + (exp(x) - 1)^k) / (1 - (exp(x) - 1)^k))^k.

%I A306081 #11 Mar 08 2024 12:00:35
%S A306081 1,2,14,134,1574,22262,367814,6907574,144942854,3357588662,
%T A306081 85000841414,2331998188214,68862337593734,2176283210561462,
%U A306081 73250933670041414,2614843434740912054,98632371931151518214,3918608865052986708662,163507638190268814991814
%N A306081 Expansion of e.g.f. Product_{k>=1} ((1 + (exp(x) - 1)^k) / (1 - (exp(x) - 1)^k))^k.
%C A306081 Convolution of A306080 and A306046.
%H A306081 Vaclav Kotesovec, <a href="/A306081/b306081.txt">Table of n, a(n) for n = 0..400</a>
%F A306081 a(n) = Sum_{k=0..n} Stirling2(n,k) * A156616(k) * k!.
%F A306081 a(n) ~ n! * exp(3 * (7*Zeta(3))^(1/3) * n^(2/3) / (4 * log(2)^(2/3)) + (1 - log(2)) * (7*Zeta(3))^(2/3) * n^(1/3) / (8 * log(2)^(4/3)) - 7*(log(2)^2 + log(2) - 1) * Zeta(3) / (48 * log(2)^2) + 1/12) * (7*Zeta(3))^(7/36) / (A * 2^(13/12) * 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 A306081 nmax = 20; CoefficientList[Series[Product[((1 + (Exp[x] - 1)^k)/(1 - (Exp[x] - 1)^k))^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
%Y A306081 Cf. A156616, A305550, A306045, A306046, A306080.
%K A306081 nonn
%O A306081 0,2
%A A306081 _Vaclav Kotesovec_, Jun 20 2018