Let $C\_n=\frac1{n+1}\binom{2n}n$ be the familiar Catalan numbers.



> 
> **QUESTION.** Is there a combinatorial or conceptual justification for this identity?
> $$\sum\_{k=1}^n\left[\frac{k}n\binom{2n}{n-k}\right]^2=C\_{2n-1}.$$
> 
> 
> 



Looking for a combinatorial proof for a Catalan identity
By the ballot theorem, $\frac{k}{n} \binom{2n}{n+k}$ is the number of Dyck paths, i.e. $(1,1), (1,-1)$-walks in the quadrant, from the origin to $(2n-1, 2k-1)$. You need to concatenate a pair of those to get a Dyck path to $(4n-2,0)$, and $k$ takes values between 1 and $n$.


