A333070 Total number of nodes summed over all 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, 2, 6, 16, 45, 132, 399, 1240, 3951, 12870, 42746, 144420, 495300, 1721202, 6051150, 21493136, 77039070, 278377452, 1013187920, 3711505380, 13675028346, 50649452084, 188482525039, 704409735912, 2642825539375, 9950643710800, 37587291143103, 142403408032648

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. A333069, A333071, A333106, 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-> (n+1)*b(n, 0):
seq(a(n), n=0..30);

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_] := (n+1) b[n, 0];
a /@ Range[0, 30] (* Jean-François Alcover, Apr 05 2021, after Alois P. Heinz *)

Formula

a(n) = (n+1) * A333069(n).

a(n) ~ c * 4^n / sqrt(n), where c = 0.0131789402414023971902275212293294628834887666310830183578424168829... - Vaclav Kotesovec, Oct 24 2021