A263144 Expansion of Product_{k>=1} 1/(1-x^(5*k-4))^k.

1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 6, 9, 9, 9, 9, 13, 19, 23, 23, 23, 28, 42, 51, 56, 56, 62, 84, 108, 120, 126, 133, 170, 219, 253, 268, 283, 335, 427, 503, 547, 574, 658, 815, 977, 1080, 1144, 1265, 1534, 1836, 2068, 2209, 2408, 2832, 3396, 3864, 4178, 4505

Offset: 0

Author

Vaclav Kotesovec, Oct 10 2015

Keywords

• nonn

Comments

In general, if s>0, t>0, GCD(s,t)=1 and g.f. = Product_{k>=1} 1/(1 - x^(s*k-t))^k then a(n) ~ s^(t^2/(3*s^2) - 7/18) * n^(t^2/(6*s^2) - 25/36) * exp(d(s,t) - Pi^4 * t^2 / (432*s^2 * Zeta(3)) + Pi^2 * t * 2^(2/3) * s^(2/3) * n^(1/3) / (12 * s^2 * Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / (2^(2/3)*s^(2/3))) / (2^(t^2/(6*s^2) + 11/36) * sqrt(3*Pi) * Zeta(3)^(t^2/(6*s^2) - 7/36)), where d(s,t) = Integral_{x=0..infinity} 1/x * (exp(-(s-t)*x)/(1 - exp(-s*x))^2 - 1/(s^2*x^2) - t/(s^2*x) + exp(-x)*(1/12 - t^2/(2*s^2))) dx.

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, A263143, A263181, A263148.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+5, 5, 'r')=1, 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-4))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; CoefficientList[Series[E^Sum[1/j*x^j/(1 - x^(5*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]

Formula

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

a(n) ~ Zeta(3)^(79/900) * exp(d54 - Pi^4/(675*Zeta(3)) + Pi^2 * 2^(2/3) * 5^(2/3) * n^(1/3) / (75*Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * 2^(-2/3) * 5^(-2/3) * n^(2/3)) / (2^(371/900) * 5^(79/450) * sqrt(3*Pi) * n^(529/900)), where d54 = A263181 = Integral_{x=0..infinity} exp(-x)/(x*(1 - exp(-5*x))^2) - 1/(25*x^3) - 4/(25*x^2) - 71/(300*x*exp(x)) = 0.1863826906247526303913683646299184833844240863417644... .