A294691 Expansion of Product_{k>=1} 1 / (1 - x^(2*k - 1))^(k*(3*k - 2)).

1, 1, 1, 9, 9, 30, 66, 106, 274, 459, 1010, 1862, 3552, 6973, 12446, 24245, 43041, 80372, 144482, 259633, 468047, 822642, 1468714, 2556542, 4493704, 7782441, 13470564, 23204471, 39679759, 67855411, 115004992, 194984378, 328183865, 551595570, 922663665

Offset: 0

Author

Vaclav Kotesovec, Nov 07 2017

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A294655, A294668, A294669.

Programs

Mathematica

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

Formula

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.