A228999 - OEIS

%I #16 Sep 13 2013 19:56:58

%S 0,1,514,21738,391026,4924020,53791050,568417200,6145964610,

%T 69860663220,844435826250,10893660591480,149983769015490,

%U 2200169585018220,34307707031519370,567195605817520080,9916409711089798530,182880725414279970660,3549284743350503607690

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

%C 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 Alois P. Heinz, <a href="/A228999/b228999.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) ~ n! * (14604*exp(1)+511)*n. - _Vaclav Kotesovec_, Sep 12 2013

%p a:= proc(n) option remember; `if`(n<4, [0, 1, 514, 21738][n+1],

%p       ((462*n^4-1028*n^3+4428*n^2-18152*n+75890)*a(n-1)

%p       -(231*n^5-745*n^4+9964*n^3-46367*n^2+184023*n-145440)*a(n-2)

%p       +2*(n-2)*(2258*n^3-10187*n^2+41200*n-31605)*a(n-3)

%p       +(n-2)*(n-3)*(231*n^3-2258*n^2+6701*n-6340)*a(n-4))/

%p       (231*n^3-514*n^2-275*n+880))

%p     end:

%p seq(a(n), n=0..30);

%t 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 Column k=9 of A229001.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 10 2013