I'm given
$$H(z)=\frac{z^4+6z}{z^6+1}$$
and I need to find $h(k)$ for $k=0,1,2,3,4$.
Where $H(z)$ is the Z transformation of $h(k)$.
Since H is very complicated, I believe some trick could be used to calculate only $h(k)$ for the specific values. Or do I need to calculate the whole inverse transformation? I could use some help either way. Thanks.
All you need to do is expand the denominator in a geometric series, and multiply by the numerator. You get
$$H(z) = z^{-6} (z^4+6 z) (1-z^{-6}+z^{-12}-\cdots) = z^{-2} + 6 z^{-5}-z^{-8}-\cdots$$
The $h[k]$ are just the coefficients of $z^{-k}$ this expansion, so you can read them right off, i.e., $h[0]=h[1]=0$, $h[2]=1$, etc.