A228995 Total sum of the 5th powers of lengths of ascending runs in all permutations of [n].

0, 1, 34, 378, 3186, 25620, 214410, 1930080, 18881250, 200907060, 2318843370, 28914797640, 387867845250, 5573855579340, 85476008430090, 1393770581296320, 24086416578328290, 439832565550644900, 8463528886854858090, 171191360282164168440, 3631513434281720800770

Offset: 0

Author

Alois P. Heinz, Sep 10 2013

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Column k=5 of A229001.

Programs

Maple

a:= proc(n) option remember; `if`(n<4, [0, 1, 34, 378][n+1],
      ((2*n^2-4*n+6)*a(n-1) -(n^3-3*n^2+13*n-10)*a(n-2)
      +2*(2*n-1)*(n-2)*a(n-3) +(n-3)*(n-2)^2*a(n-4)) /(n-2))
    end:
seq(a(n), n=0..30);

Mathematica

a[n_] := With[{k = 5}, Sum[If[n==t, 1, (n!/(t+1)!)(t(n-t+1)+1-((t+1)(n-t)+1)/(t+2))] t^k, {t, 1, n}]];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz in A229001 *)

Formula

E.g.f.: (exp(x)*(20*x^3+30*x-30)+x+30)/(x-1)^2.

a(n) ~ n! * (20*exp(1)+31)*n. - Vaclav Kotesovec, Sep 12 2013