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A288385 Expansion of Product_{k>=1} (1 - x^k)^sigma(k).

%I A288385 #24 Mar 02 2022 11:54:21
%S A288385 1,-1,-3,-1,0,10,8,12,1,-28,-29,-67,-51,-28,79,163,256,343,273,136,
%T A288385 -351,-649,-1446,-1751,-1889,-1453,-124,1924,5138,7608,10636,10903,
%U A288385 10054,3143,-5799,-20521,-37217,-53057,-65661,-66086,-54430,-15648,37179,122732
%N A288385 Expansion of Product_{k>=1} (1 - x^k)^sigma(k).
%H A288385 Seiichi Manyama, <a href="/A288385/b288385.txt">Table of n, a(n) for n = 0..1000</a>
%F A288385 Convolution inverse of A061256.
%F A288385 a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A001001(k)*a(n-k) for n > 0.
%F A288385 G.f.: exp(-Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Oct 29 2018
%p A288385 with(numtheory):
%p A288385 b:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A288385       d*sigma(d), d=divisors(j))*b(n-j), j=1..n)/n)
%p A288385     end:
%p A288385 a:= proc(n) option remember; `if`(n=0, 1,
%p A288385       -add(b(n-i)*a(i), i=0..n-1))
%p A288385     end:
%p A288385 seq(a(n), n=0..45);  # _Alois P. Heinz_, Jun 08 2017
%t A288385 b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*DivisorSigma[1, d], {d,
%t A288385      Divisors[j]}]*b[n - j], {j, 1, n}]/n];
%t A288385 a[n_] := a[n] = If[n == 0, 1, -Sum[b[n - i]*a[i], {i, 0, n - 1}]];
%t A288385 Table[a[n], {n, 0, 45}] (* _Jean-François Alcover_, Mar 02 2022, after _Alois P. Heinz_ *)
%Y A288385 Cf. A001001, A061256.
%Y A288385 Product_{k>=1} (1 - x^k)^sigma_m(k): A288098 (m=0), this sequence (m=1), A288389 (m=2), A288392 (m=3).
%K A288385 sign
%O A288385 0,3
%A A288385 _Seiichi Manyama_, Jun 08 2017