A229000 Total sum of the 10th powers of lengths of ascending runs in all permutations of [n].

0, 1, 1026, 63152, 1424406, 20708542, 247753826, 2770103322, 31016696398, 360474871982, 4422094936842, 57643276901506, 799742156488022, 11800984833241638, 184874578304981362, 3068030670168269402, 53807082887654595486, 994936476288108004702

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=10 for this sequence. - Vaclav Kotesovec, Sep 12 2013

Links

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

Crossrefs

Column k=10 of A229001.

Programs

Maple

a:= proc(n) option remember; `if`(n<5, [0, 1, 1026, 63152,
      1424406][n+1], ((398*n^3-539*n^2-4964*n+24377)*a(n-1)
      -(199*n^4+1057*n^3-12543*n^2+57436*n-31692)*a(n-2)
      +(1017*n^4-7565*n^3+34942*n^2-38827*n-17617)*a(n-3)
      -(n-3)*(1017*n^3-5258*n^2+21882*n-30370)*a(n-4)
      +(n-3)*(n-4)*(199*n^2-1212*n+1877)*a(n-5))/
      (199*n^2-778*n+792))
    end:
seq(a(n), n=0..30);

Mathematica

k=10; 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! * (83342*exp(1)-1023)*n. - Vaclav Kotesovec, Sep 12 2013