A225042 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 U=(1,1), D=(1,-1), H=(1,0) and S=(0,1).

1, 2, 8, 48, 360, 3088, 28928, 288208, 3003952, 32402384, 359019952, 4064452272, 46829600704, 547498996736, 6480275672192, 77511461858592, 935562094075392, 11381614588917296, 139425068741674448, 1718444636265140992, 21295889048851102176, 265200380258393530896

Offset: 0

Author

Alois P. Heinz, Apr 25 2013

Keywords

• nonn

Examples

			a(0) = 1: the empty path.
a(1) = 2: U, HS.
a(2) = 8: UU, HSU, UHS, HSHS, HUS, HHSS, UDSS, HSDSS.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..890

Crossrefs

Cf. A006318 (without D-steps), A224769 (without H-steps), A224776 (without U-steps), A225041 (paths to (n,0)), A286765.

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-1, y-1)+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 - 1, y - 1] + b[x, y - 1], 0] + b[x - 1, y + 1]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 29 2017, translated from Maple *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 1/6*(19009+153*sqrt(17))^(1/3) + 356/(3*(19009+153*sqrt(17))^(1/3)) + 14/3 = 13.56165398271839628518..., c = 0.03237684690282108810066870410351693504744294274892020985727414558915214336... - Vaclav Kotesovec, Sep 07 2014, updated Sep 13 2021