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

0, 1, 514, 21738, 391026, 4924020, 53791050, 568417200, 6145964610, 69860663220, 844435826250, 10893660591480, 149983769015490, 2200169585018220, 34307707031519370, 567195605817520080, 9916409711089798530, 182880725414279970660, 3549284743350503607690

Offset: 0

Author

Alois P. Heinz, Sep 10 2013

Keywords

• nonn

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Column k=9 of A229001.

Programs

Maple

a:= proc(n) option remember; `if`(n<4, [0, 1, 514, 21738][n+1],
      ((462*n^4-1028*n^3+4428*n^2-18152*n+75890)*a(n-1)
      -(231*n^5-745*n^4+9964*n^3-46367*n^2+184023*n-145440)*a(n-2)
      +2*(n-2)*(2258*n^3-10187*n^2+41200*n-31605)*a(n-3)
      +(n-2)*(n-3)*(231*n^3-2258*n^2+6701*n-6340)*a(n-4))/
      (231*n^3-514*n^2-275*n+880))
    end:
seq(a(n), n=0..30);

Mathematica

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 *)

Formula

a(n) ~ n! * (14604*exp(1)+511)*n. - Vaclav Kotesovec, Sep 12 2013