Suppose $X$ has a Poisson distribution with mean (and therefore variance) $\lambda$. Using Excel to explore properties of the distribution of $X^2$ with some small integer values of $\lambda$ I found:
The values of $E[X]$ are consistent with the formula (which was given in an answer to this question):
$$E[X^2] = \lambda^2 + \lambda$$
Trying to find a formula to fit the values of $Var[X^2]$, I came up with:
$$Var[X^2] = 4\lambda^3+6\lambda^2+\lambda$$
Is this formula generally valid, and if so how can it be proved?