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

%I A228998 #17 Sep 13 2013 19:56:10
%S A228998 0,1,258,7592,110310,1217374,12263090,123349746,1293790126,
%T A228998 14422297646,172035525354,2198386222330,30052681253126,
%U A228998 438421632024006,6806217982912546,112117997189378354,1954283594806071390,35949546988844228446,696172911589097791706
%N A228998 Total sum of the 8th powers of lengths of ascending runs in all permutations of [n].
%C A228998 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
%H A228998 Alois P. Heinz, <a href="/A228998/b228998.txt">Table of n, a(n) for n = 0..200</a>
%F A228998 a(n) ~ n! * (2914*exp(1)-255)*n. - _Vaclav Kotesovec_, Sep 12 2013
%p A228998 a:= proc(n) option remember; `if`(n<3, [0, 1, 258][n+1],
%p A228998       ((56*n^7-644*n^6+3332*n^5-9590*n^4+16016*n^3-14588*n^2
%p A228998        +5546*n+127)*a(n-1) -(n-1)*(28*n^7-280*n^6+1414*n^5
%p A228998        -4060*n^4+6748*n^3-5992*n^2+2017*n+254)*a(n-2) +(n-1)*(n-2)*
%p A228998       (28*n^6-168*n^5+490*n^4-840*n^3+868*n^2-504*n+127)*a(n-3))/
%p A228998       (28*n^6-336*n^5+1750*n^4-5040*n^3+8428*n^2-7728*n+3025))
%p A228998     end:
%p A228998 seq(a(n), n=0..30);
%t A228998 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 *)
%Y A228998 Column k=8 of A229001.
%K A228998 nonn
%O A228998 0,3
%A A228998 _Alois P. Heinz_, Sep 10 2013