A263145 Expansion of Product_{k>=1} (1+x^(5*k-1))^k.

1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 2, 3, 0, 0, 0, 4, 4, 0, 0, 1, 10, 5, 0, 0, 6, 16, 6, 0, 0, 14, 28, 7, 0, 3, 32, 40, 8, 0, 10, 63, 60, 9, 0, 33, 112, 80, 10, 3, 74, 187, 110, 11, 14, 161, 300, 140, 12, 46, 308, 455, 182, 14, 120, 568, 672, 224, 26, 283

Offset

0,10

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

Formula

G.f.: exp(Sum_{j>=1} (-1)^(j+1)/j*x^(4*j)/(1 - x^(5*j))^2).

a(n) ~ 2^(27/100) * 3^(2/3) * 5^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(32400*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(1/3) * 5^(2/3) * n^(1/3) / (900*Zeta(3)^(1/3)) + Zeta(3)^(1/3) * 3^(4/3) * 2^(2/3) * 5^(1/3) * n^(2/3) / 20) / (30 * sqrt(Pi) * n^(2/3)).

Mathematica

nmax = 100; CoefficientList[Series[Product[(1+x^(5k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(4*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A263146, A263147, A263148, A263141, A263140, A262878, A263138.

Sequence in context: A175070 A343873 A054923 * A057108 A349435 A063958

Adjacent sequences: A263142 A263143 A263144 * A263146 A263147 A263148

Keyword

nonn

Author

Vaclav Kotesovec, Oct 10 2015

Status

approved