%I #13 Jul 14 2017 06:50:35

%S 1,1,31,1341,59917,2665884,117939506,5201391077,229151753951,

%T 10097407871079,445314691051823,19662213285986440,869281482750346782,

%U 38482251447081815180,1705762097183926444500,75702251155478791228341,3363573441149092994645423

%N Number of Dyck paths of semilength 3*n and height n.

%H Alois P. Heinz, <a href="/A289473/b289473.txt">Table of n, a(n) for n = 0..604</a>

%F a(n) ~ 3^(6*n + 1/2) / (2^(4*n + 9/2) * sqrt(Pi*n)). - _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(6*n, 0, n)-b(6*n, 0, n-1)):

%p seq(a(n), n=0..20);

%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[6n, 0, n] - b[6n, 0, n - 1]]; Table[a[n], {n, 0, 20}] (* _Indranil Ghosh_, Jul 08 2017 *)

%Y Column k=3 of A289481.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jul 06 2017