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.


Duarte, Rui, and António Guedes de Oliveira. 2012. “New Developments of an Old Identity.” https://doi.org/10.48550/ARXIV.1203.5424.
Jensen, J. L. W. V. 1902. “Sur Une Identité D’abel et Sur D’autres Formules Analogues.” Acta Mathematica 26 (0): 307–18. https://doi.org/10.1007/bf02415499.
Shijie, Zhu. 1303. Jade Mirror of the Four Unknowns.