A263146 Expansion of Product_{k>=1} (1+x^(5*k-2))^k.

1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 0, 3, 0, 0, 4, 0, 4, 1, 0, 10, 0, 5, 6, 0, 16, 0, 6, 14, 0, 28, 3, 7, 32, 0, 40, 10, 8, 63, 0, 60, 33, 9, 112, 3, 80, 74, 10, 187, 14, 110, 161, 11, 300, 46, 140, 308, 13, 455, 120, 182, 568, 25, 672, 283, 224, 968, 55, 963

Offset

0,9

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^(3*j)/(1 - x^(5*j))^2).

a(n) ~ 2^(33/100) * 3^(2/3) * 5^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(8100*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(1/3) * 5^(2/3) * n^(1/3) / (450*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-2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(3*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A263145, A263147, A263148, A263142, A262879.

Sequence in context: A261202 A334596 A291900 * A365047 A361015 A028959

Adjacent sequences: A263143 A263144 A263145 * A263147 A263148 A263149

Keyword

nonn

Author

Vaclav Kotesovec, Oct 10 2015

Status

approved