up-down-right lattice paths

[Leroy Quet]

>
>1. Does anyone know anything about the sequence A_n defined by the number of
>lattice
>paths in  the plane from (0,0) to (n,n) where a step is allowed to be to the
>right, up, or
>down, but we do not allow steps to the left? No retracing allowed.
>
>2. I think it begins  2, 9, 47?, . . .
>
>3. If we change the rules to allow steps right, left, or up I believe is is
>the same count.


If you are asking what I believe you are asking, I assume then that the 
lattice path is  bounded by a square boundry, the square having opposing 
corners at (0,0) and (n,n).

(For if the lattice is unbounded, the number of paths would be infinite.)
;/

So, I believe then that the sequence is just (n+1)^n. (?)

(Since we have, with each path, simply n ways to choose from (n+1) 
vertexes independently.
(Pick from vertex (0,k) to (n,k),  1<=  k <= n.)

If I am correct, then the 47 in your sequence should be a 64.

You must be right about the same count after rotating path, assuming 
SQUARE boundry.

thanks,
Leroy Quet