A286761 Total number of nodes summed over all lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y and consist of steps D=(1,-1), H=(1,0) and S=(0,1).

1, 2, 7, 25, 106, 470, 2218, 10799, 54158, 277089, 1441956, 7602630, 40524952, 217954222, 1181107568, 6441519814, 35323986620, 194629681327, 1076819450324, 5979314763974, 33308210757892, 186074808452033, 1042146006514656, 5850075202736100, 32907053660222560

Offset: 0

Author

Alois P. Heinz, May 14 2017

Keywords

• nonn

Links

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

Crossrefs

Cf. A114296.

Programs

Maple

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

Mathematica

b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, {1, 1}, Function[
     p, p + {0, p[[1]]}][b[x - 1, y] + b[x, y - 1] + b[x - 1, y + 1]]]];
a[n_] := b[n, 0][[2]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 28 2022, after Alois P. Heinz *)