A285446 Expansion of Product_{k>=1} ((1 + x^k) / (1 - x^(3*k)))^k.

1, 1, 2, 6, 9, 18, 36, 60, 105, 191, 314, 528, 896, 1447, 2355, 3831, 6071, 9619, 15207, 23648, 36693, 56724, 86762, 132264, 200853, 302699, 454565, 680061, 1011540, 1499363, 2214570, 3255796, 4770830, 6967967, 10137577, 14703909, 21262751, 30644816, 44041843

Offset: 0

Author

Vaclav Kotesovec, Apr 19 2017

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A285445, A285447.

Cf. A156616, A000219, A285458, A285459.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^(3*k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(1/12 + 2^(-4/3) * (93*Zeta(3))^(1/3) * n^(2/3)) * (31*Zeta(3))^(7/36) / (A * 2^(7/9) * 3^(29/36) * sqrt(Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.