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

%I A228995 #15 Dec 20 2020 07:34:06
%S A228995 0,1,34,378,3186,25620,214410,1930080,18881250,200907060,2318843370,
%T A228995 28914797640,387867845250,5573855579340,85476008430090,
%U A228995 1393770581296320,24086416578328290,439832565550644900,8463528886854858090,171191360282164168440,3631513434281720800770
%N A228995 Total sum of the 5th powers of lengths of ascending runs in all permutations of [n].
%H A228995 Alois P. Heinz, <a href="/A228995/b228995.txt">Table of n, a(n) for n = 0..200</a>
%F A228995 E.g.f.: (exp(x)*(20*x^3+30*x-30)+x+30)/(x-1)^2.
%F A228995 a(n) ~ n! * (20*exp(1)+31)*n. - _Vaclav Kotesovec_, Sep 12 2013
%p A228995 a:= proc(n) option remember; `if`(n<4, [0, 1, 34, 378][n+1],
%p A228995       ((2*n^2-4*n+6)*a(n-1) -(n^3-3*n^2+13*n-10)*a(n-2)
%p A228995       +2*(2*n-1)*(n-2)*a(n-3) +(n-3)*(n-2)^2*a(n-4)) /(n-2))
%p A228995     end:
%p A228995 seq(a(n), n=0..30);
%t A228995 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}]];
%t A228995 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 20 2020, after _Alois P. Heinz_ in A229001 *)
%Y A228995 Column k=5 of A229001.
%K A228995 nonn
%O A228995 0,3
%A A228995 _Alois P. Heinz_, Sep 10 2013