login
Expansion of Product_{k>=1} 1 / (1 - x^(2*k - 1))^(k*(3*k - 2)).
3

%I #6 Nov 07 2017 09:42:24

%S 1,1,1,9,9,30,66,106,274,459,1010,1862,3552,6973,12446,24245,43041,

%T 80372,144482,259633,468047,822642,1468714,2556542,4493704,7782441,

%U 13470564,23204471,39679759,67855411,115004992,194984378,328183865,551595570,922663665

%N Expansion of Product_{k>=1} 1 / (1 - x^(2*k - 1))^(k*(3*k - 2)).

%H Vaclav Kotesovec, <a href="/A294691/b294691.txt">Table of n, a(n) for n = 0..5000</a>

%F a(n) ~ exp(Pi * 2^(5/4) / (3*5^(1/4)) * n^(3/4) + Zeta(3) * sqrt(5*n) / (Pi^2 * sqrt(2)) - (5*Zeta(3)^2 / (2*Pi^5) + Pi/24) * (5*n/2)^(1/4) + 25*Zeta(3)^3 / (3*Pi^8) + 2*Zeta(3) / (3*Pi^2) - 1/24) * sqrt(A) / (2^(173/96) * 5^(11/96) * Pi^(1/24) * n^(59/96)), where A is the Glaisher-Kinkelin constant A074962.

%t nmax = 40; CoefficientList[Series[Product[1/(1-x^(2*k-1))^(k*(3*k-2)), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A294655, A294668, A294669.

%K nonn

%O 0,4

%A _Vaclav Kotesovec_, Nov 07 2017