A363605 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^5.

0, 1, 5, 16, 35, 76, 126, 226, 335, 531, 715, 1092, 1365, 1947, 2420, 3286, 3876, 5251, 5985, 7861, 8986, 11342, 12650, 16252, 17585, 21841, 24086, 29367, 31465, 38946, 40920, 49662, 53080, 62782, 66206, 80082, 82251, 97376, 102640, 120001, 123410, 146628

Offset: 1

Author

Seiichi Manyama, Jun 11 2023

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A013663, A032741, A065608, A069153, A363604, A363606.

Cf. A000203, A001157, A001158, A001159.

Programs

Mathematica

a[n_] := DivisorSum[n, Binomial[# + 2, 4] &]; Array[a, 40] (* Amiram Eldar, Jul 25 2023 *)

PARI

my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^5)))

PARI

a(n) = my(f = factor(n)); (sigma(f, 4) + 2*sigma(f, 3) - sigma(f, 2) - 2*sigma(f)) / 24; \\ Amiram Eldar, Dec 30 2024

Formula

G.f.: Sum_{k>0} binomial(k+2,4) * x^k/(1 - x^k).

a(n) = Sum_{d|n} binomial(d+2,4).

From Amiram Eldar, Dec 30 2024: (Start)

a(n) = (sigma_4(n) + 2*sigma_3(n) - sigma_2(n) - 2*sigma_1(n)) / 24.

Dirichlet g.f.: zeta(s) * (zeta(s-4) + 2*zeta(s-3) - zeta(s-2) - 2*zeta(s-1)) / 24.

Sum_{k=1..n} a(k) ~ (zeta(5)/120) * n^5. (End)