A364351 Expansion of Sum_{k>0} k^2 * x^k/(1 + x^k)^3.

1, 1, 15, -6, 40, 12, 77, -60, 180, 30, 187, -120, 260, 56, 630, -376, 442, 117, 551, -340, 1218, 132, 805, -1104, 1325, 182, 1998, -672, 1276, 360, 1457, -2032, 2970, 306, 3290, -1710, 2072, 380, 4134, -3080, 2542, 672, 2795, -1672, 7830, 552, 3337, -6816, 4998, 775, 7038, -2340, 4240, 1080

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, A364343.

Cf. A000593, A048272, A309732.

Programs

Mathematica

a[n_] := DivisorSum[n, (-1)^(n/#+1) * (#+n) &] * 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^2*x^k/(1+x^k)^3))

Formula

a(n) = (n/2) * Sum_{d|n} (-1)^(n/d+1) * (d+n) = (n/2) * (A000593(n) + n * A048272(n)).