A263149 Expansion of Product_{k>=1} (1 + x^(2*k+1))^k.

1, 0, 0, 1, 0, 2, 0, 3, 2, 4, 4, 5, 10, 7, 16, 13, 28, 22, 40, 41, 63, 73, 90, 123, 143, 199, 214, 316, 343, 483, 532, 733, 848, 1099, 1305, 1644, 2029, 2448, 3067, 3657, 4643, 5443, 6892, 8107, 10224, 12031, 14974, 17798, 21941, 26190, 31867, 38381, 46300

Offset: 0

Author

Vaclav Kotesovec, Oct 10 2015

Keywords

• nonn

Links

• Vaclav Kotesovec, 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

Crossrefs

Cf. A035528, A263140, A263150, A263199.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k+1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(3*j)/(1 - x^(2*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^(3*j)/(1 - x^(2*j))^2).

a(n) ~ exp(-Pi^4/(5184*Zeta(3)) - Pi^2 * n^(1/3) / (8 * 3^(4/3) * Zeta(3)^(1/3)) + 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)/4) * Zeta(3)^(1/6) / (2^(23/24) * 3^(1/3)* sqrt(Pi) * n^(2/3)).