A Binomial Coefficient Identity Associated to a Conjecture of Beukers

Remark. This identity is easily verified using the WZ method, in a generalized form [Z] that applies when the summand is a hypergeometric term times a WZ potential function. It holds for all positive n, since it holds for n=1,2,3 (check!), and since the sequence defined by the sum satisfies a certain (homog.) third order linear recurrence equation. To find the recurrence, and its proof, download the Maple package EKHAD and the Maple program zeilWZP from http://www.math.temple.edu/~ zeilberg . Calling the quantity inside the braces c(n, k), compute the WZ pair (F,G), where F = c(n, k + 1)− c(n, k) and G = c(n + 1, k) − c(n, k). Go into Maple, and type read zeilWZP; zeilWZP(k*(n+k)!**2/k!**4/(n-k)!**2,F,G,k,n,N):