A263138 Expansion of Product_{k>=1} (1+x^(4*k-1))^k.

1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 3, 0, 0, 4, 4, 0, 1, 10, 5, 0, 6, 16, 6, 0, 14, 28, 7, 3, 32, 40, 8, 10, 63, 60, 9, 33, 112, 80, 13, 74, 187, 110, 25, 161, 300, 140, 58, 308, 455, 183, 133, 568, 672, 236, 297, 968, 963, 321, 609, 1609, 1344, 468, 1188, 2546

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. A263139, A263136, A263140, A262878, A263145.

Programs

Mathematica

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

a(n) ~ 2^(59/96) * 3^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(20736*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(2/3) * n^(1/3) / (288*Zeta(3)^(1/3)) + Zeta(3)^(1/3) * 2^(-8/3) * 3^(4/3) * n^(2/3)) / (12 * sqrt(Pi) * n^(2/3)).