A027346 Expansion of Product_{m>=1} (1 + q^m)^(3*m).

1, 3, 9, 28, 72, 183, 443, 1026, 2313, 5072, 10860, 22767, 46862, 94806, 188886, 371068, 719493, 1378449, 2611540, 4896291, 9090651, 16723930, 30501744, 55177932, 99048719, 176500572, 312330813, 549033172

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
• 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

Column k=3 of A277938.

Cf. A026007, A026011, A027906.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Product[(1+x^k)^(3*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 17 2015 *)

Formula

a(n) ~ exp(2^(-4/3) * 3^(5/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(11/12) * 3^(1/6) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Aug 17 2015

G.f.: exp(3*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 30 2018