TITLE: Looking for a combinatorial proof for a Catalan identity
QUESTION [13 upvotes]: 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?
$$\sum_{k=1}^n\left[\frac{k}n\binom{2n}{n-k}\right]^2=C_{2n-1}.$$

REPLY [12 votes]: More generally,
$$\sum_{k\ge1} \frac{k}{m}\binom{2m}{m-k}\cdot\frac{k}{n} \binom{2n}{n-k} = C_{m+n-1}.$$
This can be proved by the same reasoning as in Timothy Budd's answer.
This formula gives the LDU (or in this case, LU) factorization of the Hankel matrix for Catalan numbers $(C_{m+n-1})_{m,n\ge1}$. There is a similar formula for the Hankel matrix $(C_{m+n-2})_{m,n\ge1}$, involving the remaining ballot numbers. More generally, there are explicit formulas for LDU factorizations of Hankel matrices of moments of other orthogonal polynomials. (The Catalan numbers are moments of Chebyshev polynomials.)