A285295 Expansion of Product_{k>=1} (1 + x^(4*k))^(4*k) / (1 + x^k)^k.

1, -1, -1, -2, 5, -4, 0, -6, 26, -16, 6, -31, 93, -81, 19, -147, 310, -295, 136, -486, 1069, -940, 645, -1575, 3338, -3021, 2301, -5089, 9735, -9381, 7548, -15506, 27556, -27587, 23664, -44862, 76043, -77620, 70982, -124744, 204389, -211376, 203644, -336775

Offset

0,4

Links

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

Formula

a(n) ~ (-1)^n * exp(-1/12 + 3 * (5*Zeta(3))^(1/3) * n^(2/3) / 4) * A * (5*Zeta(3))^(5/36) / (2^(5/4) * sqrt(3*Pi) * n^(23/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 17 2017

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 + x^(4*k))^(4*k) / (1 + x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 16 2017 *)

Crossrefs

Product_{k>=1} (1 + x^(m*k))^(m*k) / (1 + x^k)^k: A284628 (m=2), A285294 (m=3), this sequence (m=4).

Cf. A285292.

Sequence in context: A199602 A324057 A106315 * A217563 A373427 A254881

Adjacent sequences: A285292 A285293 A285294 * A285296 A285297 A285298

Keyword

sign

Author

Seiichi Manyama, Apr 16 2017

Status

approved