A027906 Expansion of Product_{m>=1} (1+q^m)^(4*m).

1, 4, 14, 48, 141, 396, 1058, 2696, 6646, 15884, 36956, 83976, 186849, 407864, 875030, 1847824, 3845520, 7895872, 16010610, 32088120, 63611656, 124817444, 242560418, 467095640, 891754784, 1688619460

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

In general, if g.f. = Product_{m>=1} (1+x^m)^(t*m) and t>=1, then a(n) ~ 2^(-2/3 - t/12) * exp((3/2)^(4/3) * t^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * t^(1/6) * Zeta(3)^(1/6) / (3^(1/3) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Aug 17 2015

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. 19.

Crossrefs

Cf. A026007 (t=1), A026011 (t=2), A027346 (t=3).

Programs

Mathematica

nmax = 40; CoefficientList[Series[Product[(1+x^k)^(4*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 17 2015 *)

Formula

a(n) ~ exp(2^(-2/3) * 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(2/3) * 3^(1/3) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Aug 17 2015

G.f.: exp(4*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 30 2018