A263415 Expansion of Product_{k>=1} 1/(1-x^(3*k+5))^k.

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 1, 4, 0, 2, 5, 0, 6, 6, 1, 10, 7, 2, 19, 8, 6, 28, 10, 14, 44, 12, 28, 60, 17, 52, 86, 26, 93, 112, 46, 152, 152, 78, 243, 196, 142, 372, 264, 244, 552, 350, 422, 798, 486, 692, 1136, 680, 1125, 1582, 997, 1758

Offset: 0

Author

Vaclav Kotesovec, Oct 17 2015

Keywords

• nonn

Comments

In general, if v>0, GCD(v,3)=1 and g.f. = Product_{k>=1} 1/(1-x^(3*k+v))^k, then

a(n) ~ d3(v) * 3^(v^2/27 - 8/9) * exp(-Pi^4 * v^2 / (3888*Zeta(3)) - v * Pi^2 * n^(1/3) / (2^(4/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) * n^(v^2/54 - 25/36) / (sqrt(Pi) * 2^(v^2/54 + 11/36) * Zeta(3)^(v^2/54 - 7/36)), where

d3(v) = exp(Integral_{x=0..infinity} (exp((3-v)*x) / (exp(3*x)-1)^2 + (1/12 - v^2/18)/exp(x) - 1/(9*x^2) + v/(9*x))/x dx).

if mod(v,3)=1, then d3(v) = exp(A263031) * 2^((v+2)/6) * 3^((v+2)/18) * Pi^((v+2)/6) / (Gamma(1/3)^((v+2)/3) * A263416((v-1)/3)).

if mod(v,3)=2, then d3(v) = exp(A263030) * 2^((v+1)/6) * Pi^((v+1)/6) / (3^((v+1)/18) * Gamma(2/3)^((v+1)/3) * A263417((v-2)/3)).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Vaclav Kotesovec, Graph - The asymptotic ratio (80000 terms)

Crossrefs

Cf. A262877, A262876, A263405 (v=1), A263406 (v=2), A263414 (v=4), A263030, A263417.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; local r; `if`(n=0, 1,
       add(add(`if`(irem(d-3, 3, 'r')=2, d*r, 0)
        , d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Oct 17 2015

Mathematica

nmax = 80; CoefficientList[Series[Product[1/(1-x^(3*k+5))^k,{k,1,nmax}],{x,0,nmax}],x]
nmax = 80; CoefficientList[Series[E^Sum[x^(8*k)/(k*(1-x^(3*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{k>=1} x^(8*k)/(k*(1-x^(3*k))^2)).

a(n) ~ c * 3^(1/27) * exp(-25*Pi^4 / (3888*Zeta(3)) - 5*Pi^2 * n^(1/3) / (2^(4/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (sqrt(Pi) * 2^(83/108) * Zeta(3)^(29/108) * n^(25/108)), where c = exp(A263030) * Pi / (3^(1/3) * Gamma(2/3)^2) = 0.98365214791227284535715328899346961376609...