A262878 Expansion of Product_{k>=1} (1+x^(3*k-1))^k.

1, 0, 1, 0, 0, 2, 0, 2, 3, 0, 4, 4, 1, 10, 5, 6, 16, 6, 14, 28, 10, 32, 40, 18, 63, 60, 42, 112, 83, 84, 187, 124, 172, 300, 186, 320, 456, 302, 581, 684, 507, 982, 1004, 874, 1624, 1476, 1508, 2566, 2174, 2582, 3981, 3262, 4338, 6002, 4945, 7138, 8947, 7660

Offset: 0

Author

Vaclav Kotesovec, Oct 04 2015

Keywords

• nonn

Comments

In general, if s>0, t>0, GCD(s,t)=1 and g.f. = Product_{k>=1} (1 + x^(s*k-t))^k then a(n) ~ 2^(t^2/(2*s^2) - 3/4) * s^(2/3) * Zeta(3)^(1/6) * exp(-Pi^4 * t^2 / (1296 * s^2 * Zeta(3)) + Pi^2 * t * 2^(1/3) * 3^(2/3) * s^(2/3) * n^(1/3) / (36 * s^2 * Zeta(3)^(1/3)) + 3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / (2^(4/3) * s^(2/3)) ) / (3^(1/3) * s * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Oct 12 2015

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. A026007, A027346, A035528, A262876, A262877, A262879, A262884, A262948, A263138, A263145.

Programs

Maple

with(numtheory):
b:= n-> `if`(n<3, n-1, (p-> [0, -r, 2*r, 0, 0, 2*r+1][p]
         )(1+irem(n+3, 6, 'r'))):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*b(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 05 2015

Mathematica

nmax=100; CoefficientList[Series[Product[(1+x^(3k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax=100; CoefficientList[Series[E^Sum[(-1)^(j+1)/j*x^(2*j)/(1-x^(3j))^2,{j,1,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ exp(2^(-4/3) * 3^(2/3) * Zeta(3)^(1/3) * n^(2/3) + Pi^2 * n^(1/3) / (2^(5/3)*3^(8/3) * Zeta(3)^(1/3)) - Pi^4/(11664*Zeta(3))) * Zeta(3)^(1/6) / (2^(25/36) * 3^(2/3) * sqrt(Pi) * n^(2/3)).