A366135 Expansion of Sum_{k>=1} k^3 * x^k/(1 - x^k)^3.
1, 11, 33, 98, 140, 366, 371, 820, 936, 1550, 1397, 3276, 2288, 4102, 4650, 6696, 5066, 10413, 7049, 13860, 12306, 15422, 12443, 27480, 17825, 25246, 25650, 36652, 24824, 51900, 30287, 54096, 46266, 55862, 52150, 93366, 51356, 77710, 75738, 116200, 69782, 137172
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
a[n_] := (n * DivisorSigma[1, n] + DivisorSigma[2, n]) * n/2; Array[a, 50] (* Amiram Eldar, Dec 15 2023 *)
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PARI
a(n) = n*(n*sigma(n)+sigma(n, 2))/2;
Formula
a(n) = n * (n * sigma(n) + sigma_2(n))/2.
a(n) = Sum_{d|n} d^3 * binomial(n/d+1,2).
a(n) = Sum_{k=1..n} k*sigma_2(gcd(n,k)).
Sum_{k=1..n} a(k) ~ (Pi^2/48 + zeta(3)/8) * n^4. - Amiram Eldar, Dec 15 2023