There is a sequence, $P(n)$, which is defined only on perfect squares, i-e $$ P(n) = \left\{ \begin{array}{ll} \frac{e^{-\lambda}\lambda^n}{n!} & \quad n \in \{0,1,4,9,16,\cdots\} \\ 0 & \quad otherwise \end{array} \right. $$
$\lambda>0.$
Then how to evaluate the z-transform
$$\mathcal{Z}[P(n)]=\sum_0^{\infty}P(n)\cdot z^{-n} \quad \quad n \in \{0,1,4,9,16,\cdots\}.$$
Any suggestion please.
Note that: $P(n)$ is a squared Poisson random variable. I know how to calculate the first moment, $E[P[n]]=\sum_0^{\infty}n^2\frac{e^{-\lambda}\lambda^n}{n!}=\lambda + \lambda^2.$