A337318 Total number of nodes summed over all nonnegative lattice paths from (0,0) to (n,0) where the allowed steps at (x,y) are (h,v) with h in {1..max(1,y)} and v in {-1,0,1}.

1, 2, 6, 16, 45, 131, 393, 1218, 3887, 12736, 42707, 146113, 508610, 1796848, 6428953, 23253209, 84893617, 312435085, 1157899672, 4317354453, 16183476500, 60947573729, 230481995102, 874810511970, 3331322503398, 12723257204883, 48722782351656, 187028551724723

Offset: 0

Author

Alois P. Heinz, Oct 12 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1666
• Wikipedia, Counting lattice paths

Crossrefs

Cf. A337067, A337863.

Programs

Maple

b:= proc(x, y) option remember; `if`(x=0, [1$2],
      add(add((p-> p+[0, p[1]])(b(x-h, y-v)), h=1..
      min(x-y+v, max(1, y-v))), v=-1..min(y, 1)))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..30);

Mathematica

b[x_, y_] := b[x, y] = If[x == 0, {1, 1},
    Sum[Sum[Function[p, p + {0, p[[1]]}][b[x - h, y - v]], {h, 1,
    Min[x - y + v, Max[1, y - v]]}], {v, -1, Min[y, 1]}]];
a[n_] := b[n, 0][[2]];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 22 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * 4^n / sqrt(n), where c = 0.028711801689489498782112731663771630297082311282971968906589032765122715... - Vaclav Kotesovec, Oct 24 2020