A301542 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_4(k)).

1, 1, 18, 100, 526, 2546, 12953, 60929, 282194, 1265959, 5580958, 24057117, 101922204, 424244720, 1739362261, 7027590168, 28017627428, 110295521903, 429110693519, 1650961520518, 6285554480496, 23693047787961, 88469251486817, 327380976530282, 1201122749057307

Offset: 0

Author

Vaclav Kotesovec, Mar 23 2018

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..3315

Crossrefs

Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), A061256 (m=1), A275585 (m=2), A288391 (m=3), this sequence (m=4), A301543 (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8), A301547 (m=9).

Cf. A001159, A301548.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^DivisorSigma[4, k], {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(2^(3/2) * 3^(2/3) * Pi * (Zeta(5)/7)^(1/6) * n^(5/6)/5 + Pi * (7/Zeta(5))^(1/6) * n^(1/6) / (240 * sqrt(2) * 3^(2/3)) - 3*Zeta(5) / (8*Pi^4)) * Zeta(5)^(1/12) / (2^(3/4) * 3^(2/3) * 7^(1/12) * n^(7/12)).

G.f.: exp(Sum_{k>=1} sigma_5(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018