A333069 Number of lattice paths from (0,0) to (n,0) that do not go below the x-axis, and at (x,y) only allow steps (1,v) with v in {-1,0,1,...,y+1}.

1, 1, 2, 4, 9, 22, 57, 155, 439, 1287, 3886, 12035, 38100, 122943, 403410, 1343321, 4531710, 15465414, 53325680, 185575269, 651191826, 2302247822, 8194892393, 29350405663, 105713021575, 382717065800, 1392121894189, 5085836001166, 18654616951435, 68678029247822

Offset: 0

Author

Alois P. Heinz, Mar 06 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Wikipedia, Counting lattice paths
• Wikipedia, Motzkin number

Crossrefs

Cf. A196161, A333070, A333071, A333105, A333504.

Programs

Maple

b:= proc(x, y) option remember; `if`(x=0, 1, add(
     `if`(x+j>y, b(x-1, y-j), 0), j=-1-y..min(1, y)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..33);

Mathematica

b[x_, y_] := b[x, y] = If[x == 0, 1, Sum[If[x + j > y, b[x - 1, y - j], 0], {j, -1 - y, Min[1, y]}]];
a[n_] := b[n, 0];
a /@ Range[0, 33] (* Jean-François Alcover, Dec 19 2020, after Alois P. Heinz *)

Formula

a(n) = A196161(n) for n = 1..8.

a(n) ~ c * 4^n / n^(3/2), where c = 0.0131789402414023971902275212293294628834887666310830183578424168829... - Vaclav Kotesovec, Mar 25 2020