A185152 -id:A185152 - cp's OEIS frontend

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

Ilya Gutkovskiy, Sep 10 2019

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

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

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 *)

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

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

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

Original entry on oeis.org

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

Ilya Gutkovskiy, Sep 10 2019

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

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

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 *)

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

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

A374535 Expansion of (x/(8 * (1-x))) * d/dx(theta_3(x)^4).

Original entry on oeis.org

0, 1, 7, 19, 31, 61, 133, 189, 213, 330, 510, 642, 786, 968, 1304, 1664, 1712, 2018, 2720, 3100, 3460, 4132, 4924, 5476, 5764, 6539, 7631, 8711, 9383, 10253, 12413, 13405, 13501, 15085, 16921, 18601, 20005, 21411, 23691, 25875, 26595, 28317, 32349, 34241, 35825, 39335, 42647, 44903

Offset: 0

Seiichi Manyama, Jul 11 2024

nonn

Partial sums of A185152.

Cf. A014203, A374534.

my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, k^2*x^k/(1+(-x)^k)^2)/(1-x)))

a(n) = 1/8 * Sum_{i,j,k,l in Z and i^2 + j^2 + k^2 + l^2 <= n} i^2 + j^2 + k^2 + l^2.

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

cp's OEIS Frontend

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

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

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A374535 Expansion of (x/(8 * (1-x))) * d/dx(theta_3(x)^4).

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