Calculation of an impulse response of h[n] I am currently looking at the z-transform and am using a great youtube reference to help me, however I am struggling on some basic step. How do I get the impulse response array of h[n] = [ ... ] shown in the lecture at the following URL - https://www.youtube.com/watch?v=dEJp46SFgV4&amp;ab_channel=DavidDorran

I can do the following which I believe to be correct but do not know how to get the array out as shown in purple.
That purple array is giving the impulse response over time. You can get it directly from the difference equation. Assume initial rest, $y[-1]=0$, then write out the impulse response for $n=0, 1, 2, ...$. If you do that you will get:
\begin{align}
 h[0]&amp;=1+0 \\
 h[1] &amp;= 0 + 0.5 \\
 h[2] &amp;= 0 + 0.25 \\
 h[3] &amp;= 0 + 0.125 \\
 ...
\end{align}