A263406 Expansion of Product_{k>=1} 1/(1-x^(3*k+2))^k.

1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 3, 0, 2, 4, 1, 6, 5, 2, 10, 7, 6, 19, 9, 14, 29, 14, 28, 46, 23, 53, 66, 43, 95, 99, 76, 158, 143, 141, 256, 217, 247, 403, 326, 432, 617, 509, 720, 935, 801, 1187, 1399, 1281, 1892, 2087, 2047, 2983, 3107, 3272, 4589, 4647

Offset: 0

Author

Vaclav Kotesovec, Oct 17 2015

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A262877, A263030, A263405, A263414, A263415.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d-2, 3)=0, (d-2)/3, 0),
       d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 17 2015

Mathematica

nmax = 60; CoefficientList[Series[Product[1/(1-x^(3*k+2))^k,{k,1,nmax}],{x,0,nmax}],x]
nmax = 60; CoefficientList[Series[E^Sum[x^(5*k)/(k*(1-x^(3*k))^2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

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

a(n) ~ c * Zeta(3)^(13/108) * exp(-Pi^4/(972*Zeta(3)) - Pi^2 * n^(1/3) / (2^(1/3) * 3^(7/3) * Zeta(3)^(1/3)) + 3^(1/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (2^(41/108) * 3^(20/27) * sqrt(Pi) * n^(67/108)), where c = 3^(1/3) * Gamma(1/3) * exp(A263030) / sqrt(2*Pi) = 1.2763162741536982965216627321306598385267089489...