A265840 Expansion of Product_{k>=1} (1 + k^3*x^k).

1, 1, 8, 35, 91, 405, 1069, 3799, 8686, 36744, 86310, 235776, 686329, 1605779, 5230579, 13191702, 30608501, 73907925, 206052723, 433747560, 1324608945, 2995740974, 6973434054, 15364943439, 35816669079, 86662644756, 184871083828, 502089539734, 1098571699830

Offset: 0

Author

Vaclav Kotesovec, Dec 16 2015

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A022629, A092484, A265837, A265841, A265842.

Column k=3 of A292189.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Product[1 + k^3*x^k, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*j^(3*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018

Conjecture: log(a(n)) ~ 3*sqrt(n/2) * (log(2*n) - 2). - Vaclav Kotesovec, Dec 27 2020