Like the title says, I'm looking for a signal such that the bilateral Z-transform exists, but the unilateral transform does not exist for some arbitrary value or set of values for z. So, in more formal terms, there exists some $z_0\in \mathbb{C}$ such that the following sum converges
$$\sum_{n= -\infty}^{\infty}x[n]z_0^{-n}$$
but this sum does not
$$\sum_{n= 0}^{\infty}x[n]z_0^{-n}$$
Another way to put this question is if it is possible for the unilateral z-transform of a signal to have poles in the region of the complex plane where the bilateral z-transform for the same signal converges.