up-down-right lattice paths

[Henry Gould]

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.

4. How about if we allow right, left, up, or down? Again, no retracing
allowed.

5. Same problem as No. 1, but as with Catalan numbers, restrict the path so
it never rises
about the line  y = x.

I cannot find these in the OEIS.

Henry W. Gould
Dept. of Math.
West Virginia University, PO Box 6310
Morgantown, WV 26506-6310