A261736 Expansion of Product_{k>=1} (1 + x^(6*k))/(1 + x^k).

1, -1, 0, -1, 1, -1, 2, -2, 2, -3, 3, -3, 5, -5, 5, -7, 8, -8, 11, -12, 12, -16, 17, -18, 23, -25, 26, -32, 35, -37, 45, -49, 52, -62, 67, -72, 85, -92, 98, -114, 124, -133, 153, -166, 178, -203, 220, -236, 268, -290, 311, -350, 379, -407, 456, -493, 529

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), A261734 (m=4), A133563 (m=5), A113297 (m=7), A261735 (m=8), A261733 (m=9), A145707 (m=10).

Programs

Maple

seq(coeff(series(mul((1+x^(6*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^(6*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ (-1)^n * exp(sqrt(2*n)*Pi/3) / (2^(7/4)*sqrt(3)*n^(3/4)).