A301555 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(sigma(k)).

1, 2, 8, 22, 62, 154, 392, 914, 2136, 4776, 10544, 22626, 47982, 99538, 204100, 411714, 821130, 1616170, 3148812, 6066338, 11579954, 21893214, 41045780, 76306030, 140783060, 257789064, 468783092, 846697340, 1519599658, 2710476106, 4806507720, 8475250510

Offset: 0

Author

Vaclav Kotesovec, Mar 23 2018

Keywords

• nonn

Comments

Convolution of A061256 and A192065.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A000203, A061256, A192065.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^DivisorSigma[1, k], {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp((3*Pi)^(2/3) * (7*Zeta(3))^(1/3) * n^(2/3) / 2^(5/3) - 3^(1/3) * Pi^(4/3) * n^(1/3) / (2^(7/3) * (7*Zeta(3))^(1/3)) - 1/24 - Pi^2 / (224 * Zeta(3))) * A^(1/2) * (7*Zeta(3))^(11/72) / (2^(13/18) * 3^(47/72) * Pi^(11/72) * n^(47/72)), where A is the Glaisher-Kinkelin constant A074962.

G.f.: Product_{i>=1, j>=1} ((1 + x^(i*j))/(1 - x^(i*j)))^i. - Ilya Gutkovskiy, Aug 29 2018