A325941 Expansion of Sum_{k>=1} k * x^(2*k) / (1 + x^k)^2.

0, 1, -2, 5, -4, 4, -6, 17, -14, 6, -10, 28, -12, 8, -36, 49, -16, 13, -18, 46, -52, 12, -22, 100, -44, 14, -68, 64, -28, 24, -30, 129, -84, 18, -92, 121, -36, 20, -100, 166, -40, 32, -42, 100, -192, 24, -46, 292, -90, 31, -132, 118, -52, 40, -148, 232, -148, 30, -58, 264

Offset: 1

Author

Ilya Gutkovskiy, Sep 09 2019

Keywords

• sign

Links

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

Crossrefs

Cf. A000593, A048272, A094471, A143520.

Programs

Mathematica

nmax = 60; CoefficientList[Series[Sum[k x^(2 k)/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(-1)^(n/d) (n - d), {d, Divisors[n]}], {n, 1, 60}]

PARI

{a(n) = sumdiv(n, d, (-1)^(n/d)*(n-d))} \\ Seiichi Manyama, Sep 14 2019

Formula

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

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

a(n) = A000593(n) - n * A048272(n).