Solve this summation for a closed form

Question

$$h_n = \sum_{k=0}^n{\binom{-n}{n-k} \binom{n}{k}}$$

I've tried turing the negative to a positive using the identity but I get nonsense after. I summed them together but you get $\binom{0}{n}$. I am not sure what else to do. Thank you.

Answer 1

This sum is hypergeometric, so you can ask a computer to find a closed form for you. See the book A=B for more information. In the short term, you can ask sage for instance, and it will tell you the answer is $0$. This agrees with your computation that the answer is $\binom{0}{n} = 0$, unless $n=0$, in which case the $\binom{0}{0} = 1$. This agrees, moreover, with the answer suggested by @Gary's hint in the comments.

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I hope this helps ^_^