Evaluate the Z Transform
[My path was erroneous, I made a confusion, but kept a modified answer for one-sided $\mathcal{Z}$-transforms] 

With $\mathcal{Z}$-transforms, formulae with powers of $n$ often aim at making you handle basic operations on $\mathcal{Z}$-transforms: linearity, time-product/differentiation, integration, time-shift, see for instance: Properties of Z-Transform:

$$ \mathcal{Z}(nx[n]) = -z\frac{d}{dz}X(z)\,,$$
and
$$ \mathcal{Z}(x[n-n_0]) = z^{-n_0}X(z)\,.$$

Hint: if your case were a one-sided $\mathcal{Z}$-transform, one could have focused on derivation, and on the simple geometric series:

$$\sum_{n=0}^{+\infty}z^{-n} = \frac{z}{z-1}\,.$$

As of today, as pointed out by Matt and Robert, I could not make sense of the 
$$\sum_{n=-\infty}^{+\infty}z^{-n}\,,$$
and I can see but a zero radius of convergence.
For what it's worth, I will anyway try to add a common trick to try to derive something. One can use a stabilization with  some $0 &lt; a &lt;1$:

$$\sum_{n=-\infty}^{+\infty}a^{|n|}z^{-n}\,,$$
with the secret thought that, if $a$ is just close to one, we could get almost what we wanted. 
I may have made mistakes again, so far this yields the formal form:

$$ \frac{(1-a^2)z}{(z-a)(1-az)}\,.$$

This still have a singular ROC, but one can play with  $\mathcal{Z}$-transform properties.