A288389 Expansion of Product_{k>=1} (1 - x^k)^(sigma_2(k)).

1, -1, -5, -5, -1, 35, 66, 100, 15, -330, -841, -1591, -1468, 426, 6306, 16399, 27745, 31544, 6364, -70389, -225322, -435265, -617937, -537135, 176008, 1970213, 5150080, 9277624, 12631298, 11048049, -1884235, -34460900, -92385183, -171971785, -247790333

Offset: 0

Author

Seiichi Manyama, Jun 08 2017

Keywords

• sign

Links

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

Crossrefs

Cf. A027847.

Product_{k>=1} (1 - x^k)^sigma_m(k): A288098 (m=0), A288385 (m=1), this sequence (m=2), A288392 (m=3).

Programs

Magma

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1-q^k)^DivisorSigma(2,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[2](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..40);  # Alois P. Heinz, Jun 08 2017

Mathematica

nmax = 50; CoefficientList[Series[Product[(1-x^k)^DivisorSigma[2, k], {k, 1, nmax}], {x, 0, nmax}], x] (* G. C. Greubel, Oct 30 2018 *)

PARI

m=50; x='x+O('x^m); Vec(prod(k=1, m, (1-x^k)^sigma(k,2))) \\ G. C. Greubel, Oct 30 2018

Formula

Convolution inverse of A275585.

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A027847(k)*a(n-k) for n > 0.

G.f.: exp(-Sum_{k>=1} sigma_3(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 29 2018