A263397 Expansion of Product_{k>=1} 1/(1 - x^(2*k+9))^k.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 1, 7, 2, 8, 6, 9, 10, 10, 19, 11, 28, 13, 44, 15, 60, 20, 85, 29, 110, 44, 146, 69, 183, 111, 233, 171, 286, 262, 358, 391, 441, 568, 553, 808, 697, 1129, 898, 1543, 1174, 2080, 1563, 2766

Offset: 0

Author

Vaclav Kotesovec, Oct 16 2015

Keywords

• nonn

Comments

In general, if g.f. = Product_{k>=1} 1/(1-x^(2*k+v))^k and v>0 is odd, then a(n) ~ d2(v) * (2*n)^(v^2/24 - 25/36) * exp(-Pi^4 * v^2 / (1728*Zeta(3)) - Pi^2 * v * n^(1/3) /(3 * 2^(8/3) * Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) / (sqrt(3*Pi) * Zeta(3)^(v^2/24 - 7/36)), where Zeta(3) = A002117.

d2(v) = exp(Integral_{x=0..infinity} 1/(x*exp((v-2)*x) * (exp(2*x)- 1)^2) - (3*v^2-2)/(24*x*exp(x)) + v/(4*x^2) - 1/(4*x^3) dx).

d2(v) = 2^(v/4 - 1/12) * exp(-Zeta'(-1)/2) / Product_{j=1..(v-1)/2} (2*j-1)!!, where Zeta'(-1) = A084448 and Product_{j=1..(v-1)/2} (2*j-1)!! = A057863((v-1)/2).

d2(v) = 2^(1/12 + v/4 - v^2/8) * exp(1/12) * Pi^(v/4) / (A * G(v/2 + 1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
• 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
• Eric Weisstein's World of Mathematics, Barnes G-Function.

Crossrefs

Cf. A035528 (v=-1), A263150 (v=1), A263352 (v=3), A263395 (v=5), A263396 (v=7).

Programs

Maple

with(numtheory):
a:= proc(n) option remember; local r; `if`(n=0, 1,
       add(add(`if`(irem(d-8, 2, 'r')=1, d*r, 0)
       , d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..65);  # Alois P. Heinz, Oct 17 2015

Mathematica

nmax = 60; CoefficientList[Series[Product[1/(1 - x^(2*k+9))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(11*k)/(k*(1-x^(2*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{k>=1} x^(11*k)/(k*(1-x^(2*k))^2)).

a(n) ~ 16 * 2^(61/72) * exp(-1/24 - 3*Pi^4/(64*Zeta(3)) - 3*Pi^2 * n^(1/3) / (2^(8/3) * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) * sqrt(A) * n^(193/72) / (4725*sqrt(3*Pi) * Zeta(3)^(229/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.