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%I #7 Nov 08 2017 08:15:04
%S 1,2,2,10,18,36,86,150,326,608,1164,2230,4046,7632,13622,24868,44222,
%T 78304,138312,240138,418648,718292,1233494,2097350,3552370,5987642,
%U 10026088,16745600,27779030,45970868,75650248,124100970,202720814,329909400,535132036
%N Expansion of Product_{k>=1} ((1 + x^(2*k - 1))/(1 - x^(2*k - 1)))^(k^2).
%C Convolution of A294749 and A294750.
%C In general, if g.f. = Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k-1)))^(c2*k^2 + c1*k + c0) and c2 > 0, then a(n) ~ exp(Pi*sqrt(2) * c2^(1/4) * n^(3/4) / 3 + 7*(c1+c2) * Zeta(3) * sqrt(n) / (2*sqrt(c2) * Pi^2) + (Pi*(4*c0 + 2*c1 + c2) / (8*sqrt(2) * c2^(1/4)) - 49*(c1+c2)^2 * Zeta(3)^2 / (2^(3/2) * c2^(5/4) * Pi^5)) * n^(1/4) - (7*c0 + 21*c1/4 + c2 + 7*c0*c1/c2 + 7*c1^2/(2*c2)) * Zeta(3) / (4*Pi^2) + 22411*(c1+c2)^3 * Zeta(3)^3 / (196 * c2^2 * Pi^8) - (c1+c2)/24) * A^((c1+c2)/2) * (n^((c1+c2)/96 - 5/8) / (2^(c0/2 + (11*c1 + 5*c2)/48 + 9/4) * Pi^((c1+c2)/24) * c2^((c1+c2)/96 - 1/8))), where A is the Glaisher-Kinkelin constant A074962.
%H Vaclav Kotesovec, <a href="/A294755/b294755.txt">Table of n, a(n) for n = 0..5000</a>
%F a(n) ~ exp(sqrt(2)*Pi * n^(3/4)/3 + 7*Zeta(3) * sqrt(n) / (2*Pi^2) + (Pi / (8*sqrt(2)) - 49*Zeta(3)^2 / (2^(3/2) * Pi^5)) * n^(1/4) + 22411*Zeta(3)^3 / (196*Pi^8) - Zeta(3)/(4*Pi^2) - 1/24) * sqrt(A) / (2^(113/48) * Pi^(1/24) * n^(59/96)), where A is the Glaisher-Kinkelin constant A074962.
%t nmax = 50; CoefficientList[Series[Product[((1+x^(2*k-1))/(1-x^(2*k-1)))^(k^2), {k, 1, nmax}], {x, 0, nmax}], x]
%Y Cf. A206622, A292037, A294749, A294750.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Nov 08 2017
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