Question:
$$\sum_{k=0}^{n}\left ( -1 \right )^{k}\binom{2n}{k}\binom{2n-k}{2n-2k}=\sum_{2n}^{k=0}\binom{2n}{k}^{2}\left ( \frac{1+\sqrt{5}}{2} \right )^{2n-k}\left ( \frac{1-\sqrt{5}}{2} \right )^{k}$$
Attempt:
It looks like I need to start with a quadratic function like $x^{2}-x-1$ since the solutions of this function are $\frac{1\pm \sqrt{5}}{2}$. Is that the correct approach? How do I get the LHS using this function?