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Expansion of Product_{k>=1} ((1 + x^k) / (1 - x^(5*k)))^k.
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%I #8 Apr 19 2017 09:43:27

%S 1,1,2,5,8,17,29,51,88,150,254,416,682,1102,1765,2810,4415,6897,10704,

%T 16482,25251,38410,58120,87480,130999,195253,289612,427757,629128,

%U 921590,1344904,1955246,2832608,4089696,5885169,8442269,12073072,17214535,24475417

%N Expansion of Product_{k>=1} ((1 + x^k) / (1 - x^(5*k)))^k.

%C In general, if m >= 1 and g.f. = Product_{k>=1} ((1 + x^k) / (1 - x^(m*k)))^k, then a(n, m) ~ exp(1/12 + 3 * 2^(-4/3) * (3 + 4/m^2)^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * (3 + 4/m^2)^(7/36) * m^(1/12) * Zeta(3)^(7/36) / (A * 2^(7/9) * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.

%H Vaclav Kotesovec, <a href="/A285459/b285459.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ exp(1/12 + 3 * 2^(-4/3) * 5^(-2/3) * (79*Zeta(3))^(1/3) * n^(2/3)) * (79*Zeta(3))^(7/36) / (A * 2^(7/9) * 5^(11/36) * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.

%t nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^(5*k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A156616 (m=1), A000219 (m=2), A285446 (m=3), A285458 (m=4).

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Apr 19 2017