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A288392 Expansion of Product_{k>=1} (1 - x^k)^(sigma_3(k)).

%I A288392 #21 Sep 08 2022 08:46:19
%S A288392 1,-1,-9,-19,-9,163,573,1127,109,-7198,-27159,-58611,-50378,157532,
%T A288392 892986,2431694,4040909,1605559,-16109148,-68261139,-167737209,
%U A288392 -263590908,-109589779,934422499,3976197701,9922490735,16765911071,13022553978,-33008232762
%N A288392 Expansion of Product_{k>=1} (1 - x^k)^(sigma_3(k)).
%H A288392 Seiichi Manyama, <a href="/A288392/b288392.txt">Table of n, a(n) for n = 0..1000</a>
%F A288392 Convolution inverse of A288391.
%F A288392 a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A027848(k)*a(n-k) for n > 0.
%F A288392 G.f.: exp(-Sum_{k>=1} sigma_4(k)*x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Oct 29 2018
%p A288392 with(numtheory):
%p A288392 b:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A288392       d*sigma[3](d), d=divisors(j))*b(n-j), j=1..n)/n)
%p A288392     end:
%p A288392 a:= proc(n) option remember; `if`(n=0, 1,
%p A288392       -add(b(n-i)*a(i), i=0..n-1))
%p A288392     end:
%p A288392 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jun 08 2017
%t A288392 nmax = 30; CoefficientList[Series[Product[(1-x^k)^DivisorSigma[3, k], {k, 1, nmax}], {x, 0, nmax}], x] (* _G. C. Greubel_, Oct 30 2018 *)
%o A288392 (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
%o A288392 (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
%Y A288392 Cf. A027848, A288391.
%Y A288392 Product_{k>=1} (1 - x^k)^sigma_m(k): A288098 (m=0), A288385 (m=1), A288389 (m=2), this sequence (m=3).
%K A288392 sign
%O A288392 0,3
%A A288392 _Seiichi Manyama_, Jun 08 2017