A363608 Expansion of Sum_{k>0} x^(4*k)/(1-x^k)^5.
0, 0, 0, 1, 5, 15, 35, 71, 126, 215, 330, 511, 715, 1036, 1370, 1891, 2380, 3201, 3876, 5061, 6020, 7645, 8855, 11207, 12655, 15665, 17676, 21512, 23751, 29000, 31465, 37851, 41250, 48756, 52400, 62602, 66045, 77691, 82966, 96521, 101270, 118966, 123410, 143397
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[#, 4] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)
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PARI
my(N=50, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/(1-x^k)^5))) -
PARI
a(n) = my(f = factor(n)); (sigma(f, 4) - 6*sigma(f, 3) + 11*sigma(f, 2) - 6*sigma(f)) / 24; \\ Amiram Eldar, Dec 30 2024
Formula
G.f.: Sum_{k>0} binomial(k,4) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d,4).
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_4(n) - 6*sigma_3(n) + 11*sigma_2(n) - 6*sigma_1(n)) / 24.
Dirichlet g.f.: zeta(s) * (zeta(s-4) - 6*zeta(s-3) + 11*zeta(s-2) - 6*zeta(s-1)) / 24.
Sum_{k=1..n} a(k) ~ (zeta(5)/120) * n^5. (End)