up-down-right lattice paths

[jens at voss-ahrensburg.de]

> 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.

Leroy has answered this already, but it has brought me to a slight
modification:

It is easy to see that the condition "no retracing allowed" above is
equivalent to requiring the path to never touch a lattice point twice.
What happens if we take this equivalent requirement and loosen the
other condition by also allowing steps to the left?

I tried to count all possibilities and got

1, 2, 12, 148

as the first few terms (offset = 0) which does not seem to be in the
OEIS. Since this would be very surprising for me, my guess is that I
simply missed a few paths for the n = 3 case.

Does anyone have the correct figures (or even a formula for this
sequence)?

Regards,
Jens