A263147 Expansion of Product_{k>=1} (1+x^(5*k-3))^k.

1, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 0, 3, 0, 4, 0, 1, 4, 0, 10, 0, 6, 5, 0, 16, 0, 14, 6, 3, 28, 0, 32, 7, 10, 40, 0, 63, 8, 33, 60, 3, 112, 9, 74, 80, 14, 187, 10, 161, 110, 46, 300, 12, 308, 140, 120, 455, 24, 568, 182, 283, 672, 54, 968, 224, 594, 963, 146

Offset

0,8

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

a(n) ~ 2^(43/100) * 3^(2/3) * 5^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4/(3600*Zeta(3)) + Pi^2 * 3^(2/3) * 2^(1/3) * 5^(2/3) * n^(1/3) / (300*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-3))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(2*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A263145, A263146, A263148, A263143.

Sequence in context: A128765 A193511 A254218 * A298100 A303636 A073274

Adjacent sequences: A263144 A263145 A263146 * A263148 A263149 A263150

Keyword

nonn

Author

Vaclav Kotesovec, Oct 10 2015

Status

approved