A281683 Expansion of Product_{k>=1} (1 - x^(2*k-1))^(2*k-1)/(1 - x^(2*k))^(2*k).

1, -1, 2, -5, 10, -18, 32, -59, 106, -181, 305, -518, 867, -1418, 2301, -3724, 5966, -9448, 14862, -23263, 36165, -55802, 85609, -130732, 198574, -299941, 450946, -675153, 1006395, -1493598, 2207928, -3251926, 4771934, -6977018, 10166502, -14766512, 21379861

Offset: 0

Author

Seiichi Manyama, Apr 14 2017

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A224364, A285069, A281781.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 - x^k)^k/(1 - x^(2*k))^(4*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 09 2017 *)
nmax = 50; CoefficientList[Series[Product[1/((1 + x^k)^(4*k)*(1 - x^k)^(3*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 09 2017 *)

Formula

a(n) = (-1)^n * A224364(n).

a(n) ~ (-1)^n * exp(1/6 + 3 * 2^(-5/3) * (7*Zeta(3))^(1/3) * n^(2/3)) * (7*Zeta(3))^(2/9) / (2^(25/36) * A^2 * sqrt(3*Pi) * n^(13/18)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 09 2017