A Bijection for the Convolution of Central Binomial Coefficients

Let $A_n = {2n \\choose n }$ for $ n \\geq 0$ denote the the $n$-thelement in the axis of symmetry of the Pascal triangle. The generatingfunction for $A_n$ is $ (1-4t)^{-1/2}$, from which it follows that $ A_0 A_n + A_1 A_{n-1} + \\cdots + A_n A_0 = 4^n $. This note describes a bijective proof of this identity based on lattice paths.