A288421 Expansion of Product_{k>=1} 1/(1 + x^k)^sigma(k).

1, -1, -2, -2, 1, 5, 4, 10, 6, -5, -20, -27, -37, -32, -18, 23, 82, 128, 190, 185, 143, 43, -160, -424, -662, -968, -1058, -971, -571, 238, 1326, 2748, 4195, 5301, 5930, 5473, 3353, 55, -5346, -12106, -19421, -26603, -31950, -33248, -29344, -17469, 2343, 30966

Offset

0,3

Links

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

Formula

Convolution inverse of A192065.

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A288418(k)*a(n-k) for n > 0.

G.f.: exp(-Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 29 2018

Mathematica

nmax = 50; CoefficientList[Series[Product[1/(1+x^k)^DivisorSigma[1, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2017 *)

Prog

(PARI) m=50; x='x+O('x^m); Vec(prod(k=1, m+2, 1/(1+x^k)^sigma(k))) \\ G. C. Greubel, Oct 29 2018
(Magma) m:=50; R<q>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1+q^k)^DivisorSigma(1, k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018

Crossrefs

Cf. A192065, A288418.

Product_{k>=1} 1/(1 + x^k)^sigma_m(k): A288007 (m=0), this sequence (m=1), A288422 (m=2), A288423 (m=3).

Sequence in context: A101975 A136388 A099605 * A079218 A079220 A158068

Adjacent sequences: A288418 A288419 A288420 * A288422 A288423 A288424

Keyword

sign

Author

Seiichi Manyama, Jun 09 2017

Status

approved