A326830 Expansion of Product_{i>=2, j>=2} 1 / (1 - x^(i*j))^j.

1, 0, 0, 0, 2, 0, 5, 0, 9, 3, 17, 0, 46, 6, 68, 23, 153, 27, 297, 67, 534, 188, 978, 276, 1932, 620, 3250, 1313, 6033, 2246, 10854, 4361, 18776, 8639, 32831, 14835, 58230, 27635, 98052, 50980, 169522, 88243, 289720, 157179, 486232, 280206, 818006, 478014

Offset: 0

Author

Ilya Gutkovskiy, Oct 20 2019

Keywords

• nonn

Comments

Euler transform of A048050.

Convolution of A326830 and A002865 is A318784. - Vaclav Kotesovec, Oct 26 2019

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A000203, A001001, A001157, A048050, A061256, A318784, A326831.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; `if`(n<4, 0, sigma(n)-1-n) end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*b(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 20 2019

Mathematica

nmax = 47; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k] - k - 1), {k, 2, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[If[d == 1, 0, d (DivisorSigma[1, d] - d - 1)], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 47}]

Formula

G.f.: Product_{k>=1} 1 / (1 - x^k)^A048050(k).

G.f.: exp(Sum_{k>=1} (A001001(k) - A000203(k) - A001157(k) + 1) * x^k / k).

a(n) ~ exp(3^(2/3) * ((Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3)/2 - Pi^2 * (3/((Pi^2 - 6)*Zeta(3)))^(1/3) * n^(1/3)/4 - Pi^4 / (32*(Pi^2 - 6)*Zeta(3)) - 1/8) * A^(3/2)* (2*Pi)^(1/24) / (3^(1/8) * ((Pi^2 - 6)*Zeta(3))^(3/8) * n^(1/8)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Oct 26 2019