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Expansion of Sum_{k>0} k * x^(2*k-1) / (1 - x^(4*k-3)).
1

%I #16 Jul 08 2023 08:05:18

%S 1,1,3,1,4,1,5,3,6,1,7,1,10,4,9,1,10,3,11,5,12,1,18,1,14,6,15,3,16,1,

%T 17,10,24,1,19,1,20,10,21,1,25,1,30,9,24,5,25,3,26,13,27,1,36,1,29,11,

%U 30,3,38,6,32,12,42,1,34,1,35,18,36,1,37,5,48,20,39,1,48,3,41,15,42,1,54,1,48,19,45,10

%N Expansion of Sum_{k>0} k * x^(2*k-1) / (1 - x^(4*k-3)).

%F a(n) = (1/4) * Sum_{d | 4*n-2, d==1 (mod 4)} (d+3).

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

%t a[n_] := DivisorSum[4*n - 2, # + 3 &, Mod[#, 4] == 1 &]/4; Array[a, 100] (* _Amiram Eldar_, Jul 08 2023 *)

%o (PARI) a(n) = sumdiv(4*n-2, d, (d%4==1)*(d+3))/4;

%Y Cf. A364082, A364083.

%Y Cf. A363316.

%K nonn

%O 1,3

%A _Seiichi Manyama_, Jul 08 2023