cp's OEIS Frontend

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

%I A229000 #20 Jan 06 2022 14:37:29
%S A229000 0,1,1026,63152,1424406,20708542,247753826,2770103322,31016696398,
%T A229000 360474871982,4422094936842,57643276901506,799742156488022,
%U A229000 11800984833241638,184874578304981362,3068030670168269402,53807082887654595486,994936476288108004702
%N A229000 Total sum of the 10th powers of lengths of ascending runs in all permutations of [n].
%C A229000 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
%H A229000 Alois P. Heinz, <a href="/A229000/b229000.txt">Table of n, a(n) for n = 0..200</a>
%F A229000 a(n) ~ n! * (83342*exp(1)-1023)*n. - _Vaclav Kotesovec_, Sep 12 2013
%p A229000 a:= proc(n) option remember; `if`(n<5, [0, 1, 1026, 63152,
%p A229000       1424406][n+1], ((398*n^3-539*n^2-4964*n+24377)*a(n-1)
%p A229000       -(199*n^4+1057*n^3-12543*n^2+57436*n-31692)*a(n-2)
%p A229000       +(1017*n^4-7565*n^3+34942*n^2-38827*n-17617)*a(n-3)
%p A229000       -(n-3)*(1017*n^3-5258*n^2+21882*n-30370)*a(n-4)
%p A229000       +(n-3)*(n-4)*(199*n^2-1212*n+1877)*a(n-5))/
%p A229000       (199*n^2-778*n+792))
%p A229000     end:
%p A229000 seq(a(n), n=0..30);
%t A229000 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 *)
%Y A229000 Column k=10 of A229001.
%K A229000 nonn
%O A229000 0,3
%A A229000 _Alois P. Heinz_, Sep 10 2013