A286765 Total number of nodes summed over all lattice paths from (0,0) to (n,n) 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, 5, 36, 320, 3204, 34488, 389320, 4542784, 54298992, 660897208, 8157832672, 101824497960, 1282453483896, 16272274720064, 207749196820392, 2666235340584848, 34371222980687520, 444797703379924056, 5775424372048775480, 75210745056872493904

Offset: 0

Author

Alois P. Heinz, May 14 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..885

Crossrefs

Cf. A225042.

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$2)[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, n][[2]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 06 2023, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n / sqrt(n), where d = 1/6*(19009+153*sqrt(17))^(1/3) + 356/(3*(19009+153*sqrt(17))^(1/3)) + 14/3 = 13.561653982718396285180676888474... and c = 0.07613479032254374377532022793959758358787485106312078041310724993901032... - Vaclav Kotesovec, Sep 11 2021