A304459 Coefficient of x^n in Product_{k>=1} (1+x^k)^(n^3).

1, 1, 36, 3681, 770576, 276218900, 151479085752, 117975860569973, 123825991870849088, 168480096257782525419, 288418999876101261408100, 606652152400218992684850772, 1537897976017806908644807294656, 4624364862288125600795358272563097

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)^(n^m) is asymptotic to n^(m*n)/n!.

Crossrefs

Cf. A248882, A270913, A304445, A304460.

Programs

Mathematica

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

Formula

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