A262877 Expansion of Product_{k>=1} 1/(1-x^(3*k-2))^k.

1, 1, 1, 1, 3, 3, 3, 6, 9, 9, 13, 19, 23, 28, 42, 51, 62, 84, 108, 127, 170, 219, 261, 328, 427, 512, 632, 807, 987, 1190, 1504, 1838, 2214, 2744, 3374, 4036, 4950, 6060, 7260, 8793, 10748, 12853, 15459, 18766, 22473, 26834, 32425, 38768, 46136, 55376, 66168

Offset: 0

Author

Vaclav Kotesovec, Oct 04 2015

Keywords

• nonn

Comments

a(n) is the number of partitions of n into parts 3*k-2 of k kinds (k>=1). - Joerg Arndt, Oct 06 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
• Vaclav Kotesovec, Graph - The asymptotic ratio (70000 terms)

Crossrefs

Cf. A000219, A035528, A255610, A262876, A262878, A262879, A262883, A262947.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+3, 3, 'r')=1, r, 0), 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/(1-x^(3k-2))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax=100; CoefficientList[Series[E^Sum[1/j*x^j/(1-x^(3j))^2,{j,1,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ Zeta(3)^(13/108) * exp(d2 - 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 d2 = A263031 = Integral_{x=0..infinity} 1/x*(exp(-x)/(1 - exp(-3*x))^2 - 1/(9*x^2) - 2/(9*x) - 5*exp(-x)/36) = -0.0145374291832840336050202945022620903605414975934644413815... . - Vaclav Kotesovec, Oct 08 2015