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%I #20 Nov 28 2023 10:48:43
%S 1,-1,1,3,-4,-1,8,-5,1,4,-10,3,14,-8,-4,11,-16,-1,20,-12,8,10,-22,-5,
%T 21,-14,1,24,-28,4,32,-21,-10,16,-32,3,38,-20,14,20,-40,-8,44,-30,-4,
%U 22,-46,11,57,-21,-16,42,-52,-1,40,-40,20,28,-58,-12,62,-32,8,43,-56,10
%N Expansion of Sum_{k>=1} x^k * (1 - x^(2*k)) / (1 + x^k + x^(2*k))^2.
%H Amiram Eldar, <a href="/A326503/b326503.txt">Table of n, a(n) for n = 1..10000</a>
%H Claudia Rella, <a href="https://arxiv.org/abs/2212.10606">Resurgence, Stokes constants, and arithmetic functions in topological string theory</a>, arXiv:2212.10606 [hep-th], 2022. See pages 33 - 35.
%F a(n) = Sum_{d|n, d==1 (mod 3)} d - Sum_{d|n, d==2 (mod 3)} d.
%F a(n) = A078181(n) - A078182(n).
%F Multiplicative with a(3^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) if p == 1 (mod 3) and a(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 2 (mod 3). - _Amiram Eldar_, Nov 28 2023
%t nmax = 66; CoefficientList[Series[Sum[x^k (1 - x^(2 k))/(1 + x^k + x^(2 k))^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t Table[DivisorSum[n, # &, MemberQ[{1}, Mod[#, 3]] &] - DivisorSum[n, # &, MemberQ[{2}, Mod[#, 3]] &], {n, 1, 66}]
%t f[p_, e_] := If[Mod[p, 3] == 1, (p^(e + 1) - 1)/(p - 1), ((-p)^(e + 1) - 1)/(-p - 1)]; f[3, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 28 2023 *)
%o (PARI) a(n)={sumdiv(n, d, d*((d+1)%3-1))} \\ _Andrew Howroyd_, Sep 12 2019
%Y Cf. A002129, A050457, A078181, A078182, A078708, A162397 (Moebius transform), A326401.
%K sign,easy,mult
%O 1,4
%A _Ilya Gutkovskiy_, Sep 12 2019
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