A263143 Expansion of Product_{k>=1} 1/(1-x^(5*k-3))^k.

1, 0, 1, 0, 1, 0, 1, 2, 1, 2, 1, 2, 4, 2, 7, 2, 7, 6, 7, 12, 7, 16, 12, 16, 26, 16, 35, 22, 40, 44, 40, 68, 47, 80, 84, 86, 133, 94, 167, 146, 182, 238, 198, 314, 274, 358, 431, 386, 593, 494, 696, 754, 761, 1056, 917, 1288, 1326, 1430, 1880, 1664, 2348, 2290

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, A263142, A263144, A263180, A263147.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+5, 5, 'r')=2, 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-3))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^(2*j)/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

Formula

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

a(n) ~ Zeta(3)^(121/900) * exp(d53 - Pi^4/(1200*Zeta(3)) + Pi^2 * 2^(2/3) * 5^(2/3) * n^(1/3) / (100*Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * 2^(-2/3) * 5^(-2/3) * n^(2/3)) / (2^(329/900) * 5^(121/450) * sqrt(3*Pi) * n^(571/900)), where d53 = A263180 = Integral_{x=0..infinity} exp(-2*x)/(x*(1 - exp(-5*x))^2) - 1/(25*x^3) - 3/(25*x^2) - 29/(300*x*exp(x)) = -0.1461681349208040073620067065149936790708... .