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%I #9 Mar 27 2019 03:54:49
%S 1,1,1,1,25,361,3361,25201,166825,1383481,25879921,651816001,
%T 14450460025,280347467401,5253918022081,107822784560401,
%U 2578135250199625,69030779356572121,1953531819704493841,56903093167217522401,1689294590583626265625
%N Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^(k^2)).
%H Vaclav Kotesovec, <a href="/A306083/b306083.txt">Table of n, a(n) for n = 0..420</a>
%F a(n) = Sum_{k=0..n} Stirling2(n,k) * A033461(k) * k!.
%F a(n) ~ n! * exp(3 * (Pi/log(2))^(1/3) * ((sqrt(2) - 1) * Zeta(3/2))^(2/3) * n^(1/3) / 4) * ((sqrt(2) - 1) * Zeta(3/2) / Pi)^(1/3) / (2 * sqrt(6) * n^(5/6) * log(2)^(n + 1/6)).
%p a:=series(mul(1+(exp(x)-1)^(k^2),k=1..100),x=0,21): seq(n!*coeff(a, x, n),n=0..20); # _Paolo P. Lava_, Mar 26 2019
%t nmax = 20; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
%Y Cf. A033461, A306082, A306147.
%K nonn
%O 0,5
%A _Vaclav Kotesovec_, Jun 20 2018
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