Here's an identity, for some $k \in \mathbb{N}$:
$$\sum_{i=0}^k \binom{2i}{i}\binom{2k - 2i}{k - i} = 4^k$$
This is pretty easy to prove with generating functions: Consider $\frac{1}{1 - 4x}$. Then:
- The RHS is the coefficient of $x^k$ in the power series expansion
- If we write this as $(\frac{1}{\sqrt{1 - 4x}})(\frac{1}{\sqrt{1 - 4x}})$, the LHS is the coefficient of $x^k$
My question is - is there a 'combinatorial' proof of this? By which I mean - a proof that involves expressing both sides as counting the cardinality of a set, and showing that they're both ways of computing the same thing.
