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

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

Mathematica

a[n_] := (n * DivisorSigma[1, n] + DivisorSigma[2, n]) * n/2; Array[a, 50] (* Amiram Eldar, Dec 15 2023 *)

Prog

(PARI) a(n) = n*(n*sigma(n)+sigma(n, 2))/2;

Crossrefs

Cf. A007437, A309731, A309732.

Cf. A001158, A328259.

Cf. A000203, A001157.

Sequence in context: A163673 A212132 A027025 * A120354 A206527 A297478

Adjacent sequences: A366132 A366133 A366134 * A366136 A366137 A366138

Keyword

nonn,easy

Author

Seiichi Manyama, Oct 28 2023

Status

approved