A286760 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 U=(1,1), D=(1,-1), H=(1,0) and S=(0,1).

1, 2, 10, 42, 214, 1098, 5978, 33190, 189078, 1093490, 6414714, 38027030, 227489950, 1370980490, 8314674202, 50696630838, 310541818382, 1909850054666, 11786947172234, 72969941803662, 452976340653030, 2818815920369754, 17579546535174946, 109850944544149134

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. A176394, A225041.

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)+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] + 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 *)

Formula

a(n) ~ c * (3 + 2*sqrt(3))^n / sqrt(n), where c = 0.0889843039487036085233000284915570190371055498671732340656... - Vaclav Kotesovec, Sep 11 2021