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A288423
Expansion of Product_{k>=1} 1/(1 + x^k)^(sigma_3(k)).
4
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
0,3
LINKS
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
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 *)
PROG
(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
(Magma) m:=40; R<q>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1+q^k)^DivisorSigma(3, k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018
CROSSREFS
Product_{k>=1} 1/(1 + x^k)^sigma_m(k): A288007 (m=0), A288421 (m=1), A288422 (m=2), this sequence (m=3).
Sequence in context: A225912 A120081 A173206 * A373743 A081963 A208085
KEYWORD
sign
AUTHOR
Seiichi Manyama, Jun 09 2017
STATUS
approved