I came across the following finite sum involving (generalized) binomial coefficients:
$$ 2^q \sum_{k=0}^r \binom{r}{k} \binom{k/2}{q} (-1)^k .$$
Putting this into Mathematica gives me:
$$ (-1)^q 2^{r-q} \left( \binom{2q-r-1}{q-1} - \binom{2q-r-1}{q} \right) $$
and I'm interested in how could this solution be derived. There seems to be some binomial coefficient magic going on which I don't understand.
So far I have made very little progress, I noticed that the $2^q \binom{k/2}{q} = \frac{1}{q!} \prod_{i=0}^{q-1} (k-2i)$ -term looks a bit like a double factorial but this didn't get me very far. There also seems to be a lot of identities for sums involving $ \binom{r}{k} (-1)^k $ but I haven't found anything useful for this case.