A forgotten convolution type identity of Catalan: Two hypergeometric proofs (Q2725014)

Two hypergeometric proofs are given of the following identity involving the Catalan numbers \(c_n\): NEWLINE\[NEWLINE\sum^{n-1}_{k= 0} \Biggl[{(k+1)\over (2k+1)}\Biggr] c_k c_{n- (k+1)}= {1\over 2(2n+ 1)} \Biggl\{(n+ 1)c_n+ {2^{4n- 1}\over n(n+ 1)c_n}\Biggr\}.NEWLINE\]NEWLINE A discussion of Catalan's original 1887 proof was given by \textit{P. J. Larcombe} in a recent paper [A forgotten convolution type identity of Catalan, Util. Math. 57, 65-72 (2000; Zbl 0959.05011)].