A224776 Number of 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, 1, 3, 14, 83, 568, 4271, 34296, 288946, 2524676, 22695611, 208713400, 1955285936, 18601484936, 179267898087, 1746795785272, 17183086302528, 170427862676296, 1702621483524154, 17118538010217472, 173092651634957516, 1759113081143064184, 17959329720442879275

Offset: 0

Author

Alois P. Heinz, Apr 25 2013

Keywords

• nonn

Examples

			a(0) = 1: the empty path.
a(1) = 1: HS.
a(2) = 3: HSHS, HHSS, HSDSS.
a(3) = 14: HSHSHS, HHSSHS, HSDSSHS, HSHHSS, HHSHSS, HSDSHSS, HHHSSS, HSDHSSS, HSHDSSS, HHSDSSS, HSDSDSSS, HSHSDSS, HHSSDSS, HSDSSDSS.
		

Links

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

Crossrefs

Cf. A000108 (without D-steps), A114296 (paths to (n,0)), A225042 (with additional U-steps), A244593, A286764.

Programs

Maple

b:= proc(x, y) option remember; `if`(y>x, 0, `if`(x=0, 1,
       b(x-1, y)+`if`(y>0, b(x, y-1), 0)+b(x-1, y+1)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..25);

Mathematica

b[x_, y_] := b[x, y] = If[y > x, 0, If[x == 0, 1, b[x - 1, y] + If[y > 0, b[x, y - 1], 0] + b[x - 1, y + 1]]];
a[n_] := b[n, n];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * ((11+5*sqrt(5))/2)^n / n^(3/2), where c = 0.01403940208697420741365874329992235342402687... . - Vaclav Kotesovec, Sep 07 2014