A Combinatorial Identity

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}

• Duarte and de Oliveira (2012) has a combinatorial proof.

References