A363022 Expansion of Sum_{k>0} x^(2*k)/(1+x^k)^3.

0, 1, -3, 7, -10, 13, -21, 35, -39, 36, -55, 85, -78, 71, -118, 155, -136, 130, -171, 232, -234, 177, -253, 389, -310, 248, -390, 455, -406, 378, -465, 651, -586, 426, -626, 832, -666, 533, -822, 1040, -820, 734, -903, 1129, -1144, 783, -1081, 1637, -1197, 961, -1414, 1580, -1378

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, A363598, A363613, A363614.

Cf. A002129, A069153, A321543.

Programs

Mathematica

a[n_] := DivisorSum[n, (-1)^# * Binomial[#, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

PARI

my(N=60, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1+x^k)^3)))

PARI

a(n) = sumdiv(n, d, (-1)^d*binomial(d, 2));

Formula

G.f.: Sum_{k>0} binomial(k,2) * (-x)^k/(1 - x^k).

a(n) = Sum_{d|n} (-1)^d * binomial(d,2) = (A002129(n) - A321543(n))/2.