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A306080 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k)^k.

%I A306080 #14 Mar 11 2024 01:54:33
%S A306080 1,1,5,43,401,4651,64265,1015603,17996081,354373531,7682286425,
%T A306080 181466541763,4632985312961,127068851847211,3724903637434985,
%U A306080 116185013450349523,3840969677266089041,134113334651486325691,4930511086446971405945,190327859758408148070883
%N A306080 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k)^k.
%H A306080 Vaclav Kotesovec, <a href="/A306080/b306080.txt">Table of n, a(n) for n = 0..410</a>
%H A306080 Vaclav Kotesovec, <a href="/A306080/a306080.jpg">Graph - The asymptotic ratio</a>
%F A306080 a(n) = Sum_{k=0..n} Stirling2(n,k) * A026007(k) * k!.
%F A306080 a(n) ~ n! * exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / (4 * log(2)^(2/3)) + (1 - log(2)) * (3*Zeta(3))^(2/3) * n^(1/3) / (8 * log(2)^(4/3)) - (log(2)^2 + log(2) - 1) * Zeta(3) / (16 * log(2)^2)) * Zeta(3)^(1/6) / (2^(13/12) * 3^(1/3) * sqrt(Pi) * n^(2/3) * (log(2))^(n + 1/3)). - _Vaclav Kotesovec_, Jun 23 2018
%t A306080 nmax = 20; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
%Y A306080 Cf. A026007, A305550, A306046, A306081.
%K A306080 nonn
%O A306080 0,3
%A A306080 _Vaclav Kotesovec_, Jun 20 2018