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%I #12 Oct 17 2013 15:11:19
%S 1,4,41,602,11304,257400,6881160,211170960,7315701120,282398538240,
%T 12019910112000,559278036979200,28242651241728000,1538394175334016000,
%U 89912239244860032000,5612575361948755200000,372687441873534627840000,26231028469670851706880000
%N Number of permutations of [2n+1] in which the longest increasing run has length n+1.
%C Also the number of ascending runs of length n+1 in the permutations of [2n+1].
%H Alois P. Heinz, <a href="/A230251/b230251.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) = A008304(2*n+1,n+1) = A122843(2*n+1,n+1).
%F For n>0, a(n) = (5+6*n+4*n^2+n^3)*(2*n+1)!/(n+3)!. - _Vaclav Kotesovec_, Oct 15 2013
%p a:= proc(n) option remember; `if`(n<2, 1+3*n,
%p 2*n*(2*n+1)*(n^3+4*n^2+6*n+5)*a(n-1)/((n+3)*(n^3+n^2+n+2)))
%p end:
%p seq(a(n), n=0..25);
%t Flatten[{1,Table[(5+6*n+4*n^2+n^3)*(2*n+1)!/(n+3)!,{n,1,20}]}] (* _Vaclav Kotesovec_, Oct 15 2013 *)
%Y Diagonal of A008304, A122843.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Oct 13 2013
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