A363598 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^4.

0, 1, -4, 11, -20, 32, -56, 95, -124, 146, -220, 328, -364, 400, -584, 775, -816, 881, -1140, 1486, -1600, 1552, -2024, 2712, -2620, 2562, -3400, 4064, -4060, 4112, -4960, 6231, -6208, 5730, -7216, 8947, -8436, 8000, -10248, 12230, -11480, 11232, -13244, 15752

Offset: 1

Author

Seiichi Manyama, Jun 11 2023

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A325940, A363022, A363613, A363614.

Cf. A002129, A138503, A363604.

Programs

Mathematica

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

PARI

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

PARI

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

Formula

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

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