Z-transform of alternating sequence
I would approach this by first thinking of this as an upsampled version of a different signal, x(m) where m=2n, which has values at every integer.

$$x(m) = ma^{|m|/2} = 2na^{|n|}$$

So x(m=2) has the same value as x(n=4) since m=2n and x(m) is non-alternating

Then all you have to do is find the z transform of the non-alternating sequence x(m), G(z) and use the upsampling property of the z-transform(http://web.stanford.edu/class/archive/ee/ee264/ee264.1052/upsampling.pdf) to know that if:

$$x(m) \rightarrow G(z)$$
$$x(n) \rightarrow F(z)=G(z^{2})$$

If my explanation isn't sufficient let me know and I'll try to help more.