Some notes regarding the identity \begin{equation} \sum_{k=0}^n \binom{2k}k \binom{2n-2k}{n-k} = 4^n \end{equation} Gould has two derivations: The first, from Jensens equality, (18) in (Jensen 1902; Shijie 1303). A second via the Chu-Vandermonde convolution: \begin{equation} \sum_{k=0}^n \binom{x}k \binom{y}{n-k} = \binom{x+y}n \end{equation} using \(x=y=-\frac 12\) and using the $-\frac 12$-transform: \begin{equation} \binom{-1/2}{n} = (-1)^n\binom{2n}{n}\frac 1 {2^{2n}} \end{equation} Duart...
