A263139 Expansion of Product_{k>=1} (1+x^(4*k-3))^k.

1, 1, 0, 0, 0, 2, 2, 0, 0, 3, 4, 1, 0, 4, 10, 6, 0, 5, 16, 14, 3, 6, 28, 32, 10, 7, 40, 63, 33, 11, 60, 112, 74, 23, 80, 187, 161, 56, 111, 300, 308, 131, 152, 455, 568, 295, 223, 672, 968, 607, 356, 967, 1609, 1186, 618, 1367, 2546, 2189, 1132, 1926, 3941

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. A263138, A263137, A263147.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Product[(1+x^(4k-3))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^j/(1 - x^(4*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^j/(1 - x^(4*j))^2).

a(n) ~ 2^(83/96) * 3^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(2304*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(2/3) * n^(1/3) / (96*Zeta(3)^(1/3)) + Zeta(3)^(1/3) * 2^(-8/3) * 3^(4/3) * n^(2/3)) / (12 * sqrt(Pi) * n^(2/3)).