A288423 Expansion of Product_{k>=1} 1/(1 + x^k)^(sigma_3(k)).

1, -1, -8, -20, -8, 134, 512, 1062, 406, -5319, -22532, -51843, -58869, 83035, 648412, 1947384, 3665081, 3040131, -8272126, -46481039, -128400098, -234847560, -215189896, 378947363, 2437661943, 7036096665, 13868464378, 16886982518, -4042283985, -93095770772

Offset: 0

Author

Seiichi Manyama, Jun 09 2017

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..7039

Crossrefs

Cf. A288415, A288420.

Product_{k>=1} 1/(1 + x^k)^sigma_m(k): A288007 (m=0), A288421 (m=1), A288422 (m=2), this sequence (m=3).

Programs

Magma

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1+q^k)^DivisorSigma(3,k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

Maple

with(numtheory): seq(coeff(series(mul(1/(1+x^k)^(sigma[3](k)),k=1..n),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 31 2018

Mathematica

nmax = 40; CoefficientList[Series[Product[1/(1+x^k)^DivisorSigma[3, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2017 *)

PARI

m=40; x='x+O('x^m); Vec(prod(k=1, m, 1/(1+x^k)^sigma(k,3))) \\ G. C. Greubel, Oct 30 2018

Formula

Convolution inverse of A288415.

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A288420(k)*a(n-k) for n > 0.

G.f.: exp(-Sum_{k>=1} sigma_4(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 29 2018