Z-Transforms and Difference Equations?

[anyfoo]

Z-Transforms are a great tool for dealing with difference equations, for example:

y(n) = 2*x(n) + 3*x(n-1) + 4*y(n) + 2*y(n-1)

However, I don't know how to type such a difference equation into the CAS. If I just try to define it as a function, the CAS warns me that I have a recursive definition, and goes into an infinite loop if I try to actually evaluate the z-transform.

Somewhat relatedly, I noticed that simply taking the Z Transform of the Dirac function does not lead to a simple "1", but instead to

sum(z^-n * Dirac(n),n,0,inf)

Which is just the direct application of the Z Transform, without any simplification.