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Expansion of Sum_{k>0} x^(4*k) / (1 + x^(5*k)).
3

%I #10 Jul 03 2023 11:46:02

%S 0,0,0,1,0,0,0,1,-1,0,0,1,0,1,0,1,0,-1,-1,1,0,0,0,2,0,0,-1,2,-1,0,0,1,

%T 0,1,0,0,0,-1,-1,1,0,1,0,2,-1,0,0,2,-1,0,0,1,0,0,0,2,-1,-1,-1,1,0,0,

%U -1,2,0,0,0,2,-1,1,0,1,0,1,0,0,0,-1,-1,1,-1,0,0,3,0,0,-1,2,-1,-1,0,1,0,1,-1,2,0,0,-2

%N Expansion of Sum_{k>0} x^(4*k) / (1 + x^(5*k)).

%F G.f.: Sum_{k>0} (-1)^(k-1) * x^(5*k-1) / (1 - x^(5*k-1)).

%F a(n) = Sum_{d|n, d==4 (mod 5)} (-1)^d.

%t a[n_] := DivisorSum[n, (-1)^# &, Mod[#, 5] == 4 &]; Array[a, 100] (* _Amiram Eldar_, Jul 03 2023 *)

%o (PARI) a(n) = sumdiv(n, d, (d%5==4)*(-1)^d);

%Y Cf. A364043, A364044, A364045.

%Y Cf. A001899, A364022.

%K sign

%O 1,24

%A _Seiichi Manyama_, Jul 03 2023