Is there a simpler expression for the following sum? $(n\in\mathbb{N})$ $$S_n =\binom{n}{0}\binom{n}{1} + \binom{n}{1}\binom{n}{2} + \dots + \binom{n}{n-1}\binom{n}{n}$$ It seems like $S_n = \binom{2n}{n-1}$, however I have no clue as to how I can prove that relation. I also tried re-writing the sum as $$S_n = \binom{n}{0}\binom{n}{n-1} + \binom{n}{1}\binom{n}{n-2} + \dots + \binom{n}{n-1}\binom{n}{0} = \sum^{n-1}_{j=0}\binom{n}{j}\binom{n}{n-j-1}$$ Which resembles a special case of Vandemonde's Identity. Is there a connection between the two?
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See also Find a simple formula for $\binom{n}{0}\binom{n}{1}+\binom{n}{1}\binom{n}{2}+...+\binom{n}{n-1}\binom{n}{n}$. Found using Approach0. – Martin Sleziak Jan 07 '17 at 20:18