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Expansion of Sum_{k>0} k * x^k / (1 + x^(4*k)).
2

%I #20 Jul 03 2023 00:59:16

%S 1,2,3,4,4,6,7,8,10,8,11,12,12,14,12,16,18,20,19,16,20,22,23,24,21,24,

%T 30,28,28,24,31,32,34,36,28,40,36,38,36,32,42,40,43,44,40,46,47,48,50,

%U 42,54,48,52,60,44,56,58,56,59,48,60,62,67,64,48,68,67,72,68,56,71,80,74,72,63,76,76,72,79

%N Expansion of Sum_{k>0} k * x^k / (1 + x^(4*k)).

%F G.f.: Sum_{k>0} (-1)^(k-1) * x^(4*k-3) / (1 - x^(4*k-3))^2.

%F a(n) = Sum_{d|n, d==1 (mod 4)} (-1)^((d-1)/4) * (n/d).

%t a[n_] := DivisorSum[n, (-1)^((# - 1)/4) * n/# &, Mod[#, 4] == 1 &]; Array[a, 100] (* _Amiram Eldar_, Jul 03 2023 *)

%o (PARI) a(n) = sumdiv(n, d, (d%4==1)*(-1)^((d-1)/4)*n/d);

%Y Cf. A050469, A364012, A364019.

%Y Cf. A363037.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Jul 01 2023