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A228995
Total sum of the 5th powers of lengths of ascending runs in all permutations of [n].
3
0, 1, 34, 378, 3186, 25620, 214410, 1930080, 18881250, 200907060, 2318843370, 28914797640, 387867845250, 5573855579340, 85476008430090, 1393770581296320, 24086416578328290, 439832565550644900, 8463528886854858090, 171191360282164168440, 3631513434281720800770
OFFSET
0,3
LINKS
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
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 *)
CROSSREFS
Column k=5 of A229001.
Sequence in context: A202413 A168122 A271756 * A135243 A283227 A033914
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 10 2013
STATUS
approved