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Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^4.
6

%I #26 Jul 25 2023 17:18:51

%S 0,1,-4,11,-20,32,-56,95,-124,146,-220,328,-364,400,-584,775,-816,881,

%T -1140,1486,-1600,1552,-2024,2712,-2620,2562,-3400,4064,-4060,4112,

%U -4960,6231,-6208,5730,-7216,8947,-8436,8000,-10248,12230,-11480,11232,-13244,15752

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

%H Seiichi Manyama, <a href="/A363598/b363598.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) = Sum_{d|n} (-1)^d * binomial(d+1,3) = (A002129(n) - A138503(n))/6.

%t a[n_] := DivisorSum[n, (-1)^# * Binomial[# + 1, 3] &]; Array[a, 50] (* _Amiram Eldar_, Jul 25 2023 *)

%o (PARI) my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1+x^k)^4)))

%o (PARI) a(n) = sumdiv(n, d, (-1)^d*binomial(d+1, 3));

%Y Cf. A325940, A363022, A363613, A363614.

%Y Cf. A002129, A138503, A363604.

%K sign

%O 1,3

%A _Seiichi Manyama_, Jun 11 2023