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A288392
Expansion of Product_{k>=1} (1 - x^k)^(sigma_3(k)).
6
1, -1, -9, -19, -9, 163, 573, 1127, 109, -7198, -27159, -58611, -50378, 157532, 892986, 2431694, 4040909, 1605559, -16109148, -68261139, -167737209, -263590908, -109589779, 934422499, 3976197701, 9922490735, 16765911071, 13022553978, -33008232762
OFFSET
0,3
LINKS
FORMULA
Convolution inverse of A288391.
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A027848(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma_4(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 29 2018
MAPLE
with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
d*sigma[3](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..30); # Alois P. Heinz, Jun 08 2017
MATHEMATICA
nmax = 30; CoefficientList[Series[Product[(1-x^k)^DivisorSigma[3, k], {k, 1, nmax}], {x, 0, nmax}], x] (* G. C. Greubel, Oct 30 2018 *)
PROG
(PARI) m=30; x='x+O('x^m); Vec(prod(k=1, m, (1-x^k)^sigma(k, 3))) \\ G. C. Greubel, Oct 30 2018
(Magma) m:=30; R<q>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1-q^k)^DivisorSigma(3, k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018
CROSSREFS
Product_{k>=1} (1 - x^k)^sigma_m(k): A288098 (m=0), A288385 (m=1), A288389 (m=2), this sequence (m=3).
Sequence in context: A071587 A061750 A107780 * A157140 A244538 A242332
KEYWORD
sign
AUTHOR
Seiichi Manyama, Jun 08 2017
STATUS
approved