A304461 Coefficient of x^n in Product_{k>=1} ((1+x^k)/(1-x^k))^(n^3).

1, 2, 144, 29232, 12263552, 8807437800, 9671073636672, 15075101792958592, 31660212257148109824, 86182291753025176234602, 295133367252867736074882400, 1241742977667269060006125955952, 6296492342467004634980003629748736, 37869525230334631809014462278624137096

Offset: 0

Author

Vaclav Kotesovec, May 13 2018

Keywords

• nonn

Comments

In general, for m>=3, coefficient of x^n in Product_{k>=1} ((1+x^k)/(1-x^k))^(n^m) is asymptotic to 2^n * n^(m*n) / n!.

Crossrefs

Cf. A206623, A270919, A304448, A304459, A304460.

Programs

Mathematica

nmax = 20; Table[SeriesCoefficient[Product[((1+x^k)/(1-x^k))^(n^3), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 20; Table[SeriesCoefficient[(QPochhammer[-1, x]/2/QPochhammer[x])^(n^3), {x, 0, n}], {n, 0, nmax}]

Formula

a(n) ~ 2^(n - 1/2) * exp(n) * n^(2*n - 1/2) / sqrt(Pi).