A288392 Expansion of Product_{k>=1} (1 - x^k)^(sigma_3(k)).

1, -1, -9, -19, -9, 163, 573, 1127, 109, -7198, -27159, -58611, -50378, 157532, 892986, 2431694, 4040909, 1605559, -16109148, -68261139, -167737209, -263590908, -109589779, 934422499, 3976197701, 9922490735, 16765911071, 13022553978, -33008232762

Offset: 0

Author

Seiichi Manyama, Jun 08 2017

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A027848, A288391.

Product_{k>=1} (1 - x^k)^sigma_m(k): A288098 (m=0), A288385 (m=1), A288389 (m=2), this sequence (m=3).

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1-q^k)^DivisorSigma(3,k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

Maple

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*sigma[3](d), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(b(n-i)*a(i), i=0..n-1))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 08 2017

Mathematica

nmax = 30; CoefficientList[Series[Product[(1-x^k)^DivisorSigma[3, k], {k, 1, nmax}], {x, 0, nmax}], x] (* G. C. Greubel, Oct 30 2018 *)

PARI

m=30; x='x+O('x^m); Vec(prod(k=1, m, (1-x^k)^sigma(k,3))) \\ G. C. Greubel, Oct 30 2018

Formula

Convolution inverse of A288391.

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A027848(k)*a(n-k) for n > 0.

G.f.: exp(-Sum_{k>=1} sigma_4(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 29 2018