A263142 Expansion of Product_{k>=1} 1/(1-x^(5*k-2))^k.

1, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 2, 1, 3, 2, 1, 6, 2, 5, 6, 2, 11, 6, 7, 15, 6, 21, 15, 12, 30, 15, 34, 35, 22, 58, 35, 59, 70, 43, 108, 76, 95, 142, 85, 187, 157, 161, 263, 174, 318, 307, 274, 480, 336, 534, 583, 479, 836, 649, 879, 1068, 840, 1433, 1211

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. A263141, A263143, A263144, A263179, A263146, A262877.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+5, 5, 'r')=3, r, 0), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100); # after Alois P. Heinz

Mathematica

nmax = 100; CoefficientList[Series[Product[1/(1-x^(5k-2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(3*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{j>=1} 1/j*x^(3*j)/(1 - x^(5*j))^2).

a(n) ~ Zeta(3)^(151/900) * exp(d52 - Pi^4/(2700*Zeta(3)) + Pi^2 * 2^(2/3) * 5^(2/3) * n^(1/3) / (150 * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * 2^(-2/3) * 5^(-2/3) * n^(2/3)) / (2^(299/900) * 5^(151/450) * sqrt(3*Pi) * n^(601/900)), where d52 = A263179 = Integral_{x=0..infinity} exp(-3*x)/(x*(1 - exp(-5*x))^2) - 1/(25*x^3) - 2/(25*x^2) + 1/(300*x*exp(x)) = -0.187803021063745858976409657887070138806... .