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%I #4 Nov 21 2019 22:16:12
%S 1,-1,2,-1,1,-1,1,-1,2,0,1,-3,1,-1,3,-1,1,-1,1,-2,3,-1,1,-3,1,-1,2,0,
%T 1,-1,1,-1,2,-1,1,-2,1,-1,2,-2,1,-2,1,-1,4,-1,1,-3,1,0,2,-1,1,-1,2,-2,
%U 2,-1,1,-5,1,-1,3,-1,1,0,1,-1,2,0,1,-4,1,-1,3,-1,1,0,1,-2,2,-1,1,-3,1
%N Expansion of Sum_{k>=1} x^(k*(k + 1)/2) / (1 + x^(k*(k + 1)/2)).
%F G.f.: Sum_{k>=1} (-1)^(k + 1) * theta_2(x^(k/2)) / (2 * x^(k/8)).
%F a(n) = Sum_{d|n} (-1)^(n/d + 1) * A010054(d).
%t nmax = 85; CoefficientList[Series[Sum[x^(k (k + 1)/2)/(1 + x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t Table[Sum[(-1)^(n/d + 1) Boole[IntegerQ[Sqrt[8 d + 1]]], {d, Divisors[n]}], {n, 1, 85}]
%Y Cf. A000217, A007862, A010054, A048272, A304876, A317529.
%K sign
%O 1,3
%A _Ilya Gutkovskiy_, Nov 21 2019
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