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A228994 Total sum of the 4th powers of lengths of ascending runs in all permutations of [n].

%I A228994 #15 Dec 20 2020 07:34:49
%S A228994 0,1,18,152,1110,8254,66050,578466,5557246,58381646,667481754,
%T A228994 8262618730,110195259446,1576108225446,24075493095346,391282065519074,
%U A228994 6742907753730030,122830141805635966,2358555332361509066,47617194132209848026,1008436738991020480294
%N A228994 Total sum of the 4th powers of lengths of ascending runs in all permutations of [n].
%H A228994 Alois P. Heinz, <a href="/A228994/b228994.txt">Table of n, a(n) for n = 0..200</a>
%F A228994 E.g.f.: (exp(x)*(12*x^2-12*x+14)-x-14)/(x-1)^2.
%F A228994 a(n) ~ n! * (14*exp(1)-15)*n. - _Vaclav Kotesovec_, Sep 12 2013
%p A228994 a:= proc(n) option remember; `if`(n<3, [0, 1, 18][n+1],
%p A228994       ((12*n^3-42*n^2+38*n+7)*a(n-1)-(n-1)*(6*n^3-12*n^2+n+14)*a(n-2)
%p A228994        +(n-1)*(n-2)*(6*n^2-12*n+7)*a(n-3))/(6*n^2-24*n+25))
%p A228994     end:
%p A228994 seq(a(n), n=0..30);
%t A228994 a[n_] := With[{k = 4}, 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 A228994 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 20 2020, after _Alois P. Heinz_ in A229001 *)
%Y A228994 Column k=4 of A229001.
%K A228994 nonn
%O A228994 0,3
%A A228994 _Alois P. Heinz_, Sep 10 2013