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%I #41 Jun 21 2018 20:05:33
%S 1,-1,3,-6,11,-22,42,-74,131,-231,395,-669,1122,-1851,3029,-4915,7891,
%T -12572,19881,-31203,48657,-75391,116096,-177792,270822,-410394,
%U 618905,-929052,1388403,-2066140,3062270,-4520912,6649463,-9745072,14232278,-20716355,30057438
%N Expansion of Product_{k>=1} (1 + x^(2*k))^(2*k)/(1 + x^(2*k-1))^(2*k-1).
%H Vaclav Kotesovec, <a href="/A284474/b284474.txt">Table of n, a(n) for n = 0..2000</a>
%F a(n) ~ (-1)^n * exp(-1/12 + 3 * 2^(-5/3) * (7*Zeta(3))^(1/3) * n^(2/3)) * A * (7*Zeta(3))^(5/36) / (2^(10/9) * sqrt(3*Pi) * n^(23/36)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Apr 17 2017
%F G.f.: exp(Sum_{k>=1} (-1)^k*x^k/(k*(1 + x^k)^2)). - _Ilya Gutkovskiy_, Jun 20 2018
%t nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k))^(2*k)/(1 + x^(2*k-1))^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Apr 15 2017 *)
%Y Cf. A026011, A281781, A284467, A284628.
%K sign
%O 0,3
%A _Seiichi Manyama_, Apr 15 2017
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