A356244 - OEIS

%I #15 Jul 30 2022 14:14:09

%S 0,1,9,102,1304,20784,377286,7934693,186969913,4918785791,

%T 142381832107,4506907611825,154723950495961,5729421493899419,

%U 227586600129484543,9654927881195999544,435660032125475809618,20836109197604840372979,1052865018045922422499409

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

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

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

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

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

%t 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 *)

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

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

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

%Y Cf. A000330, A350125, A356131, A356243.

%K nonn

%O 1,3

%A _Seiichi Manyama_, Jul 30 2022