A301547 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_9(k)).

1, 1, 514, 20198, 414696, 12465714, 373679122, 9181285000, 224372879810, 5583837482767, 132433701077938, 3028947042351535, 68425900639083569, 1518510622688185301, 32936878700790531296, 701684036762210944310, 14726705417058058788172, 304326729686784847885978

Offset: 0

Author

Vaclav Kotesovec, Mar 23 2018

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..995

Crossrefs

Cf. A006171 (m=0), A061256 (m=1), A275585 (m=2), A288391 (m=3), A301542 (m=4), A301543 (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8).

Cf. A013957, A301553.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      sigma[9](d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 26 2018

Mathematica

nmax = 30; CoefficientList[Series[Product[1/(1-x^k)^DivisorSigma[9, k], {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp((11*Pi)^(10/11) * (Zeta(11)/3)^(1/11) * n^(10/11) / (2^(3/11) * 5^(10/11)) - Zeta'(-9)/2) * (5*Zeta(11)/(3*Pi))^(131/2904) / (2^(131/968) * 11^(1583/2904) * n^(1583/2904)).

G.f.: exp(Sum_{k>=1} sigma_10(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018