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Expansion of Product_{k>=1} 1/(1 - x^k)^A000593(k).
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%I #17 Oct 26 2018 16:52:28

%S 1,1,2,6,8,18,34,56,98,175,290,479,809,1293,2096,3382,5324,8378,13140,

%T 20319,31328,48098,73096,110763,167100,250365,373670,555613,821604,

%U 1210709,1777718,2598584,3786132,5498169,7954764,11473798,16499790,23650735,33806012

%N Expansion of Product_{k>=1} 1/(1 - x^k)^A000593(k).

%C Euler transform of A000593.

%H Seiichi Manyama, <a href="/A301799/b301799.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Vaclav Kotesovec)

%F a(n) ~ exp((3*Pi)^(2/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3) + 1/24) * Zeta(3)^(13/72) / (sqrt(A) * 2^(23/36) * 3^(49/72) * Pi^(13/72) * n^(49/72)), where A is the Glaisher-Kinkelin constant A074962.

%F G.f.: exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 + x^k))). - _Ilya Gutkovskiy_, Oct 26 2018

%t nmax = 40; CoefficientList[Series[Exp[Sum[Sum[DivisorSum[k, -(-1)^# k / # &] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 31 2018 *)

%Y Cf. A000593, A002131, A192065, A301800.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Mar 26 2018