A228998 Total sum of the 8th powers of lengths of ascending runs in all permutations of [n].

0, 1, 258, 7592, 110310, 1217374, 12263090, 123349746, 1293790126, 14422297646, 172035525354, 2198386222330, 30052681253126, 438421632024006, 6806217982912546, 112117997189378354, 1954283594806071390, 35949546988844228446, 696172911589097791706

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=8 for this sequence. - Vaclav Kotesovec, Sep 12 2013

Links

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

Crossrefs

Column k=8 of A229001.

Programs

Maple

a:= proc(n) option remember; `if`(n<3, [0, 1, 258][n+1],
      ((56*n^7-644*n^6+3332*n^5-9590*n^4+16016*n^3-14588*n^2
       +5546*n+127)*a(n-1) -(n-1)*(28*n^7-280*n^6+1414*n^5
       -4060*n^4+6748*n^3-5992*n^2+2017*n+254)*a(n-2) +(n-1)*(n-2)*
      (28*n^6-168*n^5+490*n^4-840*n^3+868*n^2-504*n+127)*a(n-3))/
      (28*n^6-336*n^5+1750*n^4-5040*n^3+8428*n^2-7728*n+3025))
    end:
seq(a(n), n=0..30);

Mathematica

k=8; 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! * (2914*exp(1)-255)*n. - Vaclav Kotesovec, Sep 12 2013