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A288389 Expansion of Product_{k>=1} (1 - x^k)^(sigma_2(k)).

%I A288389 #22 Sep 08 2022 08:46:19
%S A288389 1,-1,-5,-5,-1,35,66,100,15,-330,-841,-1591,-1468,426,6306,16399,
%T A288389 27745,31544,6364,-70389,-225322,-435265,-617937,-537135,176008,
%U A288389 1970213,5150080,9277624,12631298,11048049,-1884235,-34460900,-92385183,-171971785,-247790333
%N A288389 Expansion of Product_{k>=1} (1 - x^k)^(sigma_2(k)).
%H A288389 Seiichi Manyama, <a href="/A288389/b288389.txt">Table of n, a(n) for n = 0..1000</a>
%F A288389 Convolution inverse of A275585.
%F A288389 a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A027847(k)*a(n-k) for n > 0.
%F A288389 G.f.: exp(-Sum_{k>=1} sigma_3(k)*x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Oct 29 2018
%p A288389 with(numtheory):
%p A288389 b:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A288389       d*sigma[2](d), d=divisors(j))*b(n-j), j=1..n)/n)
%p A288389     end:
%p A288389 a:= proc(n) option remember; `if`(n=0, 1,
%p A288389       -add(b(n-i)*a(i), i=0..n-1))
%p A288389     end:
%p A288389 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jun 08 2017
%t A288389 nmax = 50; CoefficientList[Series[Product[(1-x^k)^DivisorSigma[2, k], {k, 1, nmax}], {x, 0, nmax}], x] (* _G. C. Greubel_, Oct 30 2018 *)
%o A288389 (PARI) m=50; x='x+O('x^m); Vec(prod(k=1, m, (1-x^k)^sigma(k,2))) \\ _G. C. Greubel_, Oct 30 2018
%o A288389 (Magma) m:=50; R<q>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1-q^k)^DivisorSigma(2,k): k in [1..m]]) )); // _G. C. Greubel_, Oct 30 2018
%Y A288389 Cf. A027847.
%Y A288389 Product_{k>=1} (1 - x^k)^sigma_m(k): A288098 (m=0), A288385 (m=1), this sequence (m=2), A288392 (m=3).
%K A288389 sign
%O A288389 0,3
%A A288389 _Seiichi Manyama_, Jun 08 2017