A Note on an American Mathematical Monthly “Gem”

In the Feb. 2011 issue of the American Mathematical Monthly, pp. 175-177, there is a “probabilistic” proof (by G. Chang and C. Xu) of the identity ∑_{i=0}^{n} \binom{2i}{i}\binom{2n−2i}{n−i} = 4^{n}, and of a “generalization”. My dear editors (and referees) of the AMM, have you ever heard of the Binomial theorem? Just extract the coefficient of x^{n} in ((1 − 4x)^{−1/2})^{2} = (1 − 4x)^{−1}, and as for the “generalization” do likewise to ((1 − 4x)^{−1/2})^{m} = (1 − 4x)^{−m/2} .

Doron Zeilberger, http://www.math.rutgers.edu/~zeilberg