A288422 Expansion of Product_{k>=1} 1/(1 + x^k)^(sigma_2(k)).

1, -1, -4, -6, 0, 24, 51, 89, 47, -152, -578, -1149, -1482, -738, 2384, 8901, 18476, 26774, 24151, -7143, -86804, -226605, -406442, -539872, -441822, 181268, 1671148, 4240334, 7618777, 10551340, 10218856, 1973258, -20190349, -61492391, -121880826

Offset: 0

Author

Seiichi Manyama, Jun 09 2017

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A288414, A288419.

Product_{k>=1} 1/(1 + x^k)^sigma_m(k): A288007 (m=0), A288421 (m=1), this sequence (m=2), A288423 (m=3).

Programs

Magma

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1+q^k)^DivisorSigma(2,k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018

Mathematica

nmax = 40; CoefficientList[Series[Product[1/(1+x^k)^DivisorSigma[2, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2017 *)

PARI

m=50; x='x+O('x^m); Vec(prod(k=1, m+2, 1/(1+x^k)^sigma(k,2))) \\ G. C. Greubel, Oct 29 2018

Formula

Convolution inverse of A288414.

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A288419(k)*a(n-k) for n > 0.

G.f.: exp(-Sum_{k>=1} sigma_3(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 29 2018