A286425 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) and S=(0,1).

1, 2, 8, 44, 285, 2028, 15338, 120960, 983108, 8172094, 69116592, 592590616, 5136777504, 44928712804, 395907022448, 3510622573064, 31296093794827, 280275392413204, 2520017580255461, 22736733105613548, 205767848345966976, 1867240544055742660

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. A224769.

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, 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);

Formula

a(n) ~ c * d^n / sqrt(n), where d = (3*(71 + 8*sqrt(2))^(1/3))/4 + 51/(4*(71 + 8*sqrt(2))^(1/3)) + 13/4 = 9.443535601593252082001105527294087383986236797... and c = 0.0201623254316291127574085659620180015446126055020315052104102916... - Vaclav Kotesovec, Sep 11 2021