Let $n$ be a positive integer greater than $1$. Show $$\sum_{k=1}^{n-1}\frac{1}{k(n-k)}\binom{2(k-1)}{k-1}\binom{2(n-k-1)}{n-k-1}=\frac{1}{n}\binom{2(n-1)}{n-1}$$
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