A261735 Expansion of Product_{k>=1} (1 + x^(8*k))/(1 + x^k).

1, -1, 0, -1, 1, -1, 1, -1, 3, -3, 2, -3, 4, -4, 4, -5, 8, -8, 7, -9, 11, -12, 12, -14, 20, -21, 19, -24, 28, -30, 31, -35, 45, -48, 47, -55, 64, -68, 71, -80, 97, -103, 104, -119, 135, -145, 152, -168, 198, -211, 216, -243, 272, -291, 307, -337, 386, -412

Offset: 0

Author

Vaclav Kotesovec, Aug 30 2015

Keywords

• sign

Comments

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k))/(1 + x^k), then a(n) ~ (-1)^n * exp(Pi*sqrt((m+2)*n/(6*m))) * (m+2)^(1/4) / (4 * (6*m)^(1/4) * n^(3/4)) if m is even and a(n) ~ (-1)^n * exp(Pi*sqrt((m-1)*n/(6*m))) * (m-1)^(1/4) / (2^(3/2) * (6*m)^(1/4) * n^(3/4)) if m is odd.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• 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), A261736 (m=6), A113297 (m=7), A261733 (m=9), A145707 (m=10).

Programs

Maple

seq(coeff(series(mul((1+x^(8*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^(8*k))/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ (-1)^n * exp(sqrt(5*n/6)*Pi/2) * 5^(1/4) / (2^(11/4)*3^(1/4)*n^(3/4)).