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A228999 Total sum of the 9th powers of lengths of ascending runs in all permutations of [n].

%I A228999 #16 Sep 13 2013 19:56:58
%S A228999 0,1,514,21738,391026,4924020,53791050,568417200,6145964610,
%T A228999 69860663220,844435826250,10893660591480,149983769015490,
%U A228999 2200169585018220,34307707031519370,567195605817520080,9916409711089798530,182880725414279970660,3549284743350503607690
%N A228999 Total sum of the 9th powers of lengths of ascending runs in all permutations of [n].
%C A228999 Generally, A(n,k) ~ n! * n * sum(t>=1, t^k*(t^2+t-1)/(t+2)!) = n! * n * ((Bell(k) - Bell(k+1) + sum(j=0..k, (-1)^j*(2^j*((2*k-j+1)/(j+1))-1) *Bell(k-j)*C(k,j)))*exp(1) - (-1)^k*(2^k-1)), where Bell(k) are Bell numbers A000110. Set k=9 for this sequence. - _Vaclav Kotesovec_, Sep 12 2013
%H A228999 Alois P. Heinz, <a href="/A228999/b228999.txt">Table of n, a(n) for n = 0..200</a>
%F A228999 a(n) ~ n! * (14604*exp(1)+511)*n. - _Vaclav Kotesovec_, Sep 12 2013
%p A228999 a:= proc(n) option remember; `if`(n<4, [0, 1, 514, 21738][n+1],
%p A228999       ((462*n^4-1028*n^3+4428*n^2-18152*n+75890)*a(n-1)
%p A228999       -(231*n^5-745*n^4+9964*n^3-46367*n^2+184023*n-145440)*a(n-2)
%p A228999       +2*(n-2)*(2258*n^3-10187*n^2+41200*n-31605)*a(n-3)
%p A228999       +(n-2)*(n-3)*(231*n^3-2258*n^2+6701*n-6340)*a(n-4))/
%p A228999       (231*n^3-514*n^2-275*n+880))
%p A228999     end:
%p A228999 seq(a(n), n=0..30);
%t A228999 k=9; Table[n^k+Sum[t^k*n!*(n*(t^2+t-1)-t*(t^2-4)+1)/(t+2)!+Floor[t/n]*(1/(t*(t+3)+2)),{t,1,n-1}],{n,0,20}] (* _Vaclav Kotesovec_, Sep 12 2013 *)
%Y A228999 Column k=9 of A229001.
%K A228999 nonn
%O A228999 0,3
%A A228999 _Alois P. Heinz_, Sep 10 2013