OFFSET
0,9
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.
FORMULA
a(n) ~ (-1)^n * exp(sqrt(5*n/6)*Pi/2) * 5^(1/4) / (2^(11/4)*3^(1/4)*n^(3/4)).
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]
CROSSREFS
KEYWORD
sign
AUTHOR
Vaclav Kotesovec, Aug 30 2015
STATUS
approved