A294667 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(3*k+1)/2).
1, 2, 10, 33, 110, 332, 997, 2829, 7889, 21299, 56400, 146028, 371681, 929498, 2290296, 5562369, 13336036, 31583177, 73957845, 171342592, 393018517, 893000610, 2011039286, 4490680381, 9947577333, 21867539862, 47721817473, 103420870299, 222641160569
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Programs
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Mathematica
nmax = 30; CoefficientList[Series[Product[1/(1-x^k)^(k*(3*k+1)/2), {k, 1, nmax}], {x, 0, nmax}], x]
Formula
a(n) ~ exp(Pi * 2^(7/4) * n^(3/4) / (3*5^(1/4)) + Zeta(3) * sqrt(5*n) / (sqrt(2) * Pi^2) - 5^(5/4) * Zeta(3)^2 * n^(1/4) / (2^(7/4) * Pi^5) + (25 * Zeta(3)^3) / (6*Pi^8) - 3*Zeta(3) / (8*Pi^2) + 1/24) * Pi^(1/24) / (sqrt(A) * 2^(157/96) * 5^(13/96) * n^(61/96)), where A is the Glaisher-Kinkelin constant A074962.