A356131 - OEIS

%I #16 Jul 28 2022 15:50:35

%S 0,1,9,100,1302,20648,377022,7921039,186926431,4916562309,

%T 142373072781,4506381442625,154721361953489,5729251983077521,

%U 227585590018322461,9654855432715969784,435659531345223039702,20836069677785611552293

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

%F a(n) = Sum_{k=1..n} k * (sigma_{n-1}(k) - floor(n/k)^n) = A356129(n) - A350109(n).

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

%F a(n) = [x^n] (1/(1-x)) * Sum_{k>=1} (k - 1)^n * x^k/(1 - x^k)^2.

%t a[n_] := Sum[(k - 1)^n * Binomial[Floor[n/k]+1, 2], {k, 1, n}]; Array[a, 18] (* _Amiram Eldar_, Jul 28 2022 *)

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

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

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

%Y Cf. A350109, A356100, A356129.

%K nonn

%O 1,3

%A _Seiichi Manyama_, Jul 27 2022