A pretty binomial identity

Elementary proofs abound: the first identity results from choosing x = y = 1 in the binomial expansion of (x+y). The second one may be obtained by comparing the coefficient of x in the identity (1 + x)(1 + x) = (1 + x). The reader is surely aware of many other proofs, including some combinatorial in nature. At the end of the previous century, the evaluation of these sums was trivialized by the work of H. Wilf and D. Zeilberger [8]. In the preface to the charming book [8], the authors begin with the phrase You’ve been up all night working on your new theory, you found the answer, and it is in the form that involves factorials, binomial coefficients, and so on, ... and then proceed to introduce the method of creative telescoping discussed in Section 3. This technique provides an automatic tool for the verification of these type of identities. The points of view presented in [3] and [10] provide an entertaining comparison of what is admissible as a proof. In this short note we present a variety of proofs of the identity