A286764 Total number of nodes summed over all lattice paths from (0,0) to (n,n) that do not go below the x-axis or above the diagonal x=y and consist of steps D=(1,-1), H=(1,0) and S=(0,1).

1, 3, 16, 109, 855, 7298, 65838, 617118, 5946781, 58506642, 584894463, 5921596628, 60565217546, 624644829720, 6487216108058, 67767838847144, 711463437534474, 7501409431304796, 79386836213817417, 842882477863610604, 8974911258934880498, 95806877080558096428

Offset: 0

Author

Alois P. Heinz, May 14 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..960

Crossrefs

Cf. A001622, A224776.

Programs

Maple

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, [1$2],
      (p-> p+[0, p[1]])(b(x-1, y)+b(x, y-1)+b(x-1, y+1))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..30);

Mathematica

b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, {1, 1}, Function[
     p, p + {0, p[[1]]}][b[x - 1, y] + b[x, y - 1] + b[x - 1, y + 1]]]];
a[n_] := b[n, n][[2]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 28 2022, after Alois P. Heinz *)

Formula

a(n) ~ c * phi^(5*n) / sqrt(n), where phi = A001622 is the golden ratio and c = 0.036755631845424682385214848270310481743236419858524834059514156934711202... - Vaclav Kotesovec, Sep 11 2021