Another Proof of an Extension of a Curious Identity

. (1) In 2002 Z. W. Sun [7] proved the case when z = 1 using double recursion. Later four alternative proofs have been provided for the special case. A generating function proof was given by A. Panholzer and H. Prodinger [6]; D. Merlini and R. Sprugnoli [5] established it through Riordan arrays; S. B. Ekhad and M. Mohammed [4] proved it based on a WZ method. Later, W. Chu and L. V. D. Claudio [3] reobtained the identity by using Jensen formulae [2]. More recently, D. Callan [1] gave a combinatorial proof using weight-reversing involutions on suitable configurations involving dominos and colorings. Motivated by the method [3], we give a simple proof of the identity (1).