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Expansion of Product_{k>=1} 1/(1 - x^k/(1 - x)^k)^k.
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%I #9 Oct 15 2018 22:19:54

%S 1,1,4,13,41,125,374,1103,3213,9259,26430,74806,210095,585890,1623240,

%T 4470232,12241799,33349751,90410255,243977941,655553258,1754265279,

%U 4676358086,12420299846,32873598566,86721264126,228051843891,597905347237,1563071037798,4074973824099

%N Expansion of Product_{k>=1} 1/(1 - x^k/(1 - x)^k)^k.

%C First differences of the binomial transform of A000219.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F G.f.: exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x)^k)).

%F a(n) ~ Zeta(3)^(7/36) * 2^(n - 11/18) * exp(3*Zeta(3)^(1/3) * n^(2/3) / 2^(4/3) + Zeta(3)^(2/3) * n^(1/3) / 2^(5/3) + (1 - Zeta(3))/12) / (A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Oct 15 2018

%p seq(coeff(series(mul((1-x^k/(1-x)^k)^(-k),k=1..n),x,n+1), x, n), n = 0 .. 29); # _Muniru A Asiru_, Oct 15 2018

%t nmax = 29; CoefficientList[Series[Product[1/(1 - x^k/(1 - x)^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 29; CoefficientList[Series[Exp[Sum[DivisorSigma[2, k] x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]

%Y Cf. A000219, A001157, A103446, A218482, A294500, A320564.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Oct 15 2018