## Is there a closed form expression for this 2D generalization of the Fibonacci sequence?

You've essentially described the Delannoy Numbers.  See http://www.research.att.com/~nja… for a list of references.

This isn't exactly a closed form, but here's a formula that might be helpful to you:

$f(x,y) = \sum_{d=0}^{\min\{x-1,y-1\}} 2^d \binom{x-1}{d}\binom{y-1}{d}.$