A224769 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) and S=(0,1).

1, 1, 2, 7, 33, 184, 1142, 7629, 53750, 394157, 2981546, 23117242, 182867360, 1470714606, 11993628444, 98967634147, 824958769631, 6937180941468, 58785077008641, 501520244718945, 4304433733010962, 37142428443486254, 322042675618484973, 2804409601249038670

Offset: 0

Author

Alois P. Heinz, Apr 17 2013

Keywords

• nonn

Examples

			a(2) = 2: UDSS, UU.
a(3) = 7: UDSDSSS, UDUSS, UDSSDSS, UUDSS, UDSUS, UDSSU, UUU.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A198324 (paths to (n,0)), A225042 (with additional H-steps), A286425.

Programs

Maple

b:= proc(x, y) option remember; `if`(y>x, 0, `if`(x=0, 1,
      `if`(y>0, b(x, y-1)+b(x-1, y-1), 0)+b(x-1, y+1)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..30);

Mathematica

b[x_, y_] := b[x, y] = If[y > x, 0, If[x == 0, 1, If[y > 0, b[x, y - 1] + b[x - 1, y - 1], 0] + b[x - 1, y + 1]]];
a[n_] := b[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 3/4*(71 + 8*sqrt(2))^(1/3) + 51/(4*(71 + 8*sqrt(2))^(1/3)) + 13/4 = 9.4435356015932520820011..., c = 0.00814413508604516738631686716788556507884786... . - Vaclav Kotesovec, Sep 07 2014