A364343 Expansion of Sum_{k>0} k * x^k/(1 + x^k)^3.

1, -1, 9, -12, 20, -12, 35, -60, 72, -30, 77, -132, 104, -56, 210, -256, 170, -117, 209, -320, 378, -132, 299, -672, 425, -182, 594, -588, 464, -360, 527, -1040, 858, -306, 910, -1224, 740, -380, 1170, -1640, 902, -672, 989, -1364, 1890, -552, 1175, -2928, 1470, -775, 1938, -1872, 1484, -1080, 2090

Offset: 1

Author

Seiichi Manyama, Jul 19 2023

Keywords

• sign

Links

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

Crossrefs

Cf. A320900, A364351.

Cf. A002129, A048272, A309731.

Programs

Mathematica

a[n_] := DivisorSum[n, (-1)^(# + 1)*(# + 1) &] * n/2; Array[a, 55] (* Amiram Eldar, Jul 20 2023 *)

PARI

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

Formula

a(n) = (n/2) * Sum_{d|n} (-1)^(d+1) * (d+1) = (n/2) * (A002129(n) + A048272(n)).