$ x[n] = \begin{cases} 1, & n = 4 \\ 2, & n = 5 \\ 3, & n = 2 \\ 0, & \mbox{else} \end{cases} $
This is equivalent to
$ \begin{align} x[n] &= u[n-4] + 2u[n-5] + 3u[n-2] \\ & {\color{blue} \text{I think you mean } \delta[n-4] + 2\delta [n-5] + 3\delta [n-2]}, \\ {\color{blue} \text{and thus } X(z)} &{\color{blue}= z^{-4} + 2z^{-5} + 3z^{-2} , \text{ which converges for any finite }z\neq 0.}\\ \end{align} $
