A262380 Expansion of Product_{k>=1} 1/((1+x^k)*(1-x^k)^4).

1, 3, 10, 25, 62, 136, 293, 590, 1165, 2205, 4097, 7391, 13120, 22780, 38997, 65613, 109036, 178660, 289575, 463842, 735870, 1155717, 1799620, 2777795, 4254859, 6467115, 9761770, 14633605, 21799465, 32273399, 47506759, 69537814, 101252595, 146675875, 211451893

Offset: 0

Author

Vaclav Kotesovec, Sep 20 2015

Keywords

• nonn

Comments

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

Links

• 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. 16.

Crossrefs

Cf. A002513 (m=2), A029863 (m=3), A261998.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[1/((1 + x^k)*(1 - x^k)^4), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(sqrt(7*n/3)*Pi) * 7^(5/4) / (32 * 3^(5/4) * n^(7/4)).