A333504 Sum of the heights of 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}.

0, 0, 1, 3, 9, 28, 88, 282, 921, 3058, 10302, 35159, 121406, 423704, 1493046, 5307276, 19014642, 68609686, 249149529, 910000728, 3341113126, 12325295866, 45664033813, 169846998495, 634020229888, 2374550269819, 8920273989351, 33604033638696, 126919824985533

Offset: 0

Author

Alois P. Heinz, Mar 24 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400
• Wikipedia, Counting lattice paths

Crossrefs

Cf. A333069, A333070, A333071, A333498, A333608.

Programs

Maple

b:= proc(x, y, h) option remember; `if`(x=0, h, add((t->
     `if`(x>t, b(x-1, t, max(h, t)), 0))(y-j), j=-1-y..min(1, y)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..33);

Mathematica

b[x_, y_, h_] := b[x, y, h] = If[x == 0, h, Sum[With[{t = y - j},
     If[x > t, b[x - 1, t, Max[h, t]], 0]], {j, -1 - y, Min[1, y]}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 33] (* Jean-François Alcover, Apr 26 2021, after Alois P. Heinz *)