A363607 Expansion of Sum_{k>0} x^(3*k)/(1-x^k)^4.

0, 0, 1, 4, 10, 21, 35, 60, 85, 130, 165, 245, 286, 399, 466, 620, 680, 921, 969, 1274, 1366, 1705, 1771, 2325, 2310, 2886, 3010, 3679, 3654, 4666, 4495, 5580, 5622, 6664, 6590, 8285, 7770, 9405, 9426, 11210, 10660, 13230, 12341, 14953, 14740, 16951, 16215, 20181

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. A013662, A069153, A363608.

Cf. A059358, A363604, A363611.

Cf. A000203, A001157, A001158.

Programs

Mathematica

a[n_] := DivisorSum[n, Binomial[#, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

PARI

my(N=50, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/(1-x^k)^4)))

PARI

a(n) = my(f = factor(n)); (sigma(f, 3) - 3*sigma(f, 2) + 2*sigma(f)) / 6; \\ Amiram Eldar, Dec 30 2024

Formula

G.f.: Sum_{k>0} binomial(k,3) * x^k/(1 - x^k).

a(n) = Sum_{d|n} binomial(d,3).

From Amiram Eldar, Dec 30 2024: (Start)

a(n) = (sigma_3(n) - 3*sigma_2(n) + 2*sigma_1(n)) / 6.

Dirichlet g.f.: zeta(s) * (zeta(s-3) - 3*zeta(s-2) + 2*zeta(s-1)) / 6.

Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)