A261734 Expansion of Product_{k>=1} (1 + x^(4*k))/(1 + x^k).

1, -1, 0, -1, 2, -2, 1, -2, 4, -4, 3, -4, 8, -8, 6, -9, 14, -14, 12, -16, 24, -25, 22, -28, 40, -42, 38, -48, 65, -68, 64, -78, 102, -108, 104, -124, 159, -168, 164, -194, 242, -256, 254, -296, 362, -385, 386, -444, 536, -570, 576, -658, 782, -832, 848, -961

Offset: 0

Author

Vaclav Kotesovec, Aug 30 2015

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• David J. Hemmer, Generating functions for fixed points of the Mullineux map, arXiv:2402.03643 [math.CO], 2024.
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 14.

Crossrefs

Cf. A081360 (m=2), A109389 (m=3), A133563 (m=5), A261736 (m=6), A113297 (m=7), A261735 (m=8), A261733 (m=9), A145707 (m=10).

Programs

Maple

seq(coeff(series(mul((1+x^(4*k))/(1+x^k),k=1..n), x,n+1),x,n),n=0..60); # Muniru A Asiru, Jul 29 2018

Mathematica

nmax = 100; CoefficientList[Series[Product[(1 + x^(4*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ (-1)^n * exp(sqrt(n)*Pi/2) / (4*sqrt(2)*n^(3/4)).