A289473 Number of Dyck paths of semilength 3*n and height n.

1, 1, 31, 1341, 59917, 2665884, 117939506, 5201391077, 229151753951, 10097407871079, 445314691051823, 19662213285986440, 869281482750346782, 38482251447081815180, 1705762097183926444500, 75702251155478791228341, 3363573441149092994645423

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..604

Formula

a(n) ~ 3^(6*n + 1/2) / (2^(4*n + 9/2) * sqrt(Pi*n)). - Vaclav Kotesovec, Jul 14 2017

Maple

b:= proc(x, y, k) option remember;
      `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))
    end:
a:= n-> `if`(n=0, 1, b(6*n, 0, n)-b(6*n, 0, n-1)):
seq(a(n), n=0..20);

Mathematica

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 *)

Crossrefs

Column k=3 of A289481.

Sequence in context: A078961 A261856 A049292 * A069406 A234115 A057008

Adjacent sequences: A289470 A289471 A289472 * A289474 A289475 A289476

Keyword

nonn

Author

Alois P. Heinz, Jul 06 2017

Status

approved