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%I #17 Sep 08 2022 08:46:24
%S 1,0,5,-5,7,0,9,-18,18,0,13,-25,15,0,35,-47,19,0,21,-35,45,0,25,-90,
%T 38,0,58,-45,31,0,33,-108,65,0,63,-90,39,0,75,-126,43,0,45,-65,126,0,
%U 49,-235,66,0,95,-75,55,0,91,-162,105,0,61,-175,63,0,162,-233,105
%N Expansion of Sum_{k>=1} tau(k) * x^k / (1 + x^k)^2, where tau = A000005.
%C Inverse Moebius transform of A002129.
%C Dirichlet convolution of A000005 with A181983.
%F a(n) = Sum_{d|n} A002129(d).
%F a(n) = Sum_{d|n} (-1)^(d + 1) * d * tau(n/d).
%t nmax = 65; CoefficientList[Series[Sum[DivisorSigma[0, k] x^k/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t a[n_] := Sum[(-1)^(d + 1) d DivisorSigma[0, n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 65}]
%o (PARI) a(n) = {sumdiv(n, d, (-1)^(d + 1) * d * numdiv(n/d))} \\ _Andrew Howroyd_, Sep 14 2019
%o (Magma) [&+[(-1)^(d+1)*d*#Divisors(n div d):d in Divisors(n)]:n in [1..65]]; // _Marius A. Burtea_, Sep 14 2019
%Y Cf. A000005, A002129, A007429, A008586 (positions of negative terms), A016825 (positions of 0's), A181983, A288417, A288571.
%K sign,mult
%O 1,3
%A _Ilya Gutkovskiy_, Sep 14 2019
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