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
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
a[n_] := DivisorSum[n, Binomial[# - 1, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)
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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
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)