A326125 Expansion of Sum_{k>=1} k^2 * x^k / (1 + x^k)^2.

1, 2, 12, 4, 30, 24, 56, 8, 117, 60, 132, 48, 182, 112, 360, 16, 306, 234, 380, 120, 672, 264, 552, 96, 775, 364, 1080, 224, 870, 720, 992, 32, 1584, 612, 1680, 468, 1406, 760, 2184, 240, 1722, 1344, 1892, 528, 3510, 1104, 2256, 192, 2793, 1550, 3672, 728, 2862, 2160

Offset: 1

Author

Ilya Gutkovskiy, Sep 10 2019

Keywords

• nonn
• mult

Links

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

Crossrefs

Cf. A000593, A064987, A078306, A143520, A185152, A245579, A326238, A353908.

Programs

Mathematica

nmax = 54; CoefficientList[Series[Sum[k^2 x^k/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[n Sum[(-1)^(n/d + 1) d, {d, Divisors[n]}], {n, 1, 54}]
f[p_, e_] := p^e*(p^(e+1)-1)/(p-1); f[2, e_] := 2^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)

PARI

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

Formula

G.f.: Sum_{k>=1} (-1)^(k + 1) * k * x^k * (1 + x^k) / (1 - x^k)^3.

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

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

From Amiram Eldar, Dec 05 2022: (Start)

Multiplicative with a(2^e) = 2^e, and a(p^e) = p^e*(p^(e+1)-1)/(p-1) for p > 2.

Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^2/36 = 0.2741556... (A353908). (End)

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