%I #11 Jul 14 2017 10:17:36

%S 1,1,7,1341,5828185,517500496981,877820839402932499,

%T 27202373147496127842409429,14934414860406931133627906259665137,

%U 142143740345412121643458345045577780672138977,23087568034858117342849941754170955046637454778184629205

%N Number of Dyck paths of semilength n^2 and height n.

%H Alois P. Heinz, <a href="/A289482/b289482.txt">Table of n, a(n) for n = 0..40</a>

%F a(n) = A289481(n,n).

%F a(n) ~ c * 2^(2*n^2) / n^4, where c = 0.034180619793706218467525729844898502557235639065782754227258170112282483988... - _Vaclav Kotesovec_, Jul 14 2017

%p b:= proc(x, y, k) option remember;

%p `if`(x=0, 1, `if`(y>0, b(x-1, y-1, k), 0)+

%p `if`(y < min(x-1, k), b(x-1, y+1, k), 0))

%p end:

%p a:= n-> `if`(n=0, 1, b(2*n^2, 0, n)-b(2*n^2, 0, n-1)):

%p seq(a(n), n=0..12);

%t b[x_, y_, k_]:=b[x, y, k]=If[x==0, 1, If[y>0, b[x - 1, y - 1, k], 0] + If[y<Min[x - 1, k], b[x - 1, y + 1, k], 0]]; a[n_]:=a[n]=If[n==0, 1, b[2n^2, 0, n] - b[2n^2, 0, n - 1]]; Table[a[n], {n, 0, 12}] (* _Indranil Ghosh_, Jul 08 2017 *)

%Y Main diagonal of A289481.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jul 06 2017