A356244 a(n) = Sum_{k=1..n} (k-1)^n * Sum_{j=1..floor(n/k)} j^2.

0, 1, 9, 102, 1304, 20784, 377286, 7934693, 186969913, 4918785791, 142381832107, 4506907611825, 154723950495961, 5729421493899419, 227586600129484543, 9654927881195999544, 435660032125475809618, 20836109197604840372979, 1052865018045922422499409

Offset: 1

Author

Seiichi Manyama, Jul 30 2022

Keywords

• nonn

Crossrefs

Cf. A000330, A350125, A356131, A356243.

Programs

Mathematica

a[n_] := Sum[(k - 1)^n * Sum[j^2, {j, 1, Floor[n/k]}], {k, 1, n}]; Array[a, 19] (* Amiram Eldar, Jul 30 2022 *)

PARI

a(n) = sum(k=1, n, (k-1)^n*sum(j=1, n\k, j^2));

PARI

a(n) = sum(k=1, n, k^2*(sigma(k, n-2)-(n\k)^n));

PARI

a(n) = sum(k=1, n, k^2*sumdiv(k, d, (d-1)^n/d^2));

Formula

a(n) = Sum_{k=1..n} (k-1)^n * A000330(floor(n/k)).

a(n) = Sum_{k=1..n} k^2 * (sigma_{n-2}(k) - floor(n/k)^n) = A356243(n) - A350125(n).

a(n) = Sum_{k=1..n} k^2 * Sum_{d|k} (d - 1)^n / d^2.

a(n) = [x^n] (1/(1-x)) * Sum_{k>=1} (k - 1)^n * x^k * (1 + x^k)/(1 - x^k)^3.