A363610 Expansion of Sum_{k>0} x^(3*k)/(1-x^k)^3.

0, 0, 1, 3, 6, 11, 15, 24, 29, 42, 45, 69, 66, 93, 98, 129, 120, 175, 153, 216, 206, 255, 231, 343, 282, 366, 354, 447, 378, 550, 435, 594, 542, 648, 582, 828, 630, 819, 770, 978, 780, 1114, 861, 1161, 1072, 1221, 1035, 1529, 1143, 1494, 1346, 1644, 1326, 1878, 1482, 1953

Offset

1,4

Links

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

Formula

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

a(n) = Sum_{d|n} binomial(d-1,2).

From Amiram Eldar, Jan 01 2025: (Start)

a(n) = (sigma_2(n) - 3*sigma_1(n) + 2*sigma_0(n)) / 2.

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

Sum_{k=1..n} a(k) ~ (zeta(3)/6) * n^3. (End)

Mathematica

a[n_] := DivisorSum[n, Binomial[# - 1, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)

Prog

(PARI) my(N=60, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/(1-x^k)^3)))
(PARI) a(n) = my(f = factor(n)); (sigma(f, 2) - 3*sigma(f) + 2*numdiv(f)) / 2; \\ Amiram Eldar, Jan 01 2025

Crossrefs

Cf. A000005, A000203, A001157, A002117, A065608, A363611.

Cf. A007437, A069153.

Sequence in context: A059753 A256001 A131665 * A132158 A363644 A363652

Adjacent sequences: A363607 A363608 A363609 * A363611 A363612 A363613

Keyword

nonn,easy

Author

Seiichi Manyama, Jun 11 2023

Status

approved