A326238 Expansion of Sum_{k>=1} k * x^k * (1 - x^k) / (1 + x^k)^3.

1, -2, 12, -20, 30, -24, 56, -104, 117, -60, 132, -240, 182, -112, 360, -464, 306, -234, 380, -600, 672, -264, 552, -1248, 775, -364, 1080, -1120, 870, -720, 992, -1952, 1584, -612, 1680, -2340, 1406, -760, 2184, -3120, 1722, -1344, 1892, -2640, 3510, -1104, 2256

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

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

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

a(n) = n * A002129(n).

Multiplicative with a(2^e) = 2^e*(3-2^(e+1)), and a(p^e) = p^e*(p^(e+1)-1)/(p-1) if p > 2. - Amiram Eldar, Dec 05 2022

Dirichlet g.f.: zeta(s-1)*zeta(s-2)*(1-2^(3-s)). - Amiram Eldar, Jan 07 2023

Mathematica

nmax = 47; CoefficientList[Series[Sum[k x^k (1 - x^k)/(1 + x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[n Sum[(-1)^(d + 1) d, {d, Divisors[n]}], {n, 1, 47}]
f[p_, e_] := p^e*(p^(e+1)-1)/(p-1); f[2, e_] := 2^e*(3-2^(e+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)

Prog

(PARI) a(n)={n*sumdiv(n, d, (-1)^(d + 1) * d)} \\ Andrew Howroyd, Sep 10 2019

Crossrefs

Cf. A002129, A064987, A113184, A143520, A185152, A326125.

Sequence in context: A136725 A023534 A286683 * A226399 A353399 A121859

Adjacent sequences: A326235 A326236 A326237 * A326239 A326240 A326241

Keyword

sign,mult

Author

Ilya Gutkovskiy, Sep 10 2019

Status

approved