A326831 Expansion of Product_{i>=2, j>=2} (1 + x^(i*j))^j.

1, 0, 0, 0, 2, 0, 5, 0, 7, 3, 17, 0, 37, 6, 58, 23, 120, 21, 235, 67, 390, 161, 726, 230, 1349, 521, 2225, 1055, 3990, 1714, 7040, 3341, 11604, 6294, 20053, 10500, 34252, 19115, 56055, 34168, 94306, 56998, 157078, 99515, 254766, 171484, 419287, 283565

Offset: 0

Author

Ilya Gutkovskiy, Oct 20 2019

Keywords

• nonn

Comments

Weigh transform of A048050.

Convolution of A326831 and A025147 is A319107. - Vaclav Kotesovec, Oct 26 2019

Links

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

Crossrefs

Cf. A048050, A192065, A319107, A326830.

Programs

Maple

with(numtheory):
g:= proc(n) option remember; `if`(n<4, 0, sigma(n)-1-n) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(g(i), j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 20 2019

Mathematica

nmax = 47; CoefficientList[Series[Product[(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, (-1)^(k/d + 1) 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 + x^k)^A048050(k).

a(n) ~ exp(3*(2*(Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3)/4 - Pi^2 * n^(1/3) / (2^(7/3) * ((Pi^2 - 6)*Zeta(3))^(1/3)) - Pi^4 / (96*(Pi^2 - 6)*Zeta(3))) * 2^(19/24) * ((Pi^2 - 6)*Zeta(3))^(1/6) / (sqrt(3*Pi) * n^(2/3)). - Vaclav Kotesovec, Oct 26 2019