A289474 Number of Dyck paths of semilength 4*n and height n.

1, 1, 127, 26609, 5828185, 1244027317, 258054207727, 52402156308977, 10488803639337560, 2079716852900452250, 409870041643305514072, 80466951297324661639450, 15759844503630649015980996, 3082231209190859509953543989, 602325527248635884295374077954

Offset: 0

Author

Alois P. Heinz, Jul 06 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400

Crossrefs

Column k=4 of A289481.

Programs

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(8*n, 0, n)-b(8*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[yIndranil Ghosh, Jul 08 2017 *)

Formula

a(n) ~ 2^(24*n + 4) / (3^(3*n + 1/2) * 5^(5*n + 7/2) * sqrt(Pi*n)). - Vaclav Kotesovec, Jul 14 2017