A colleague asked me if the following was true and I can't seem to give him an answer. Any help would be appreciated. He has a fair amount of numerical evidence, but just be a victim of the law of small numbers.
Let q be an odd prime such that n = 2q + 1 is also a prime. Let g be an even integer in the interval [2, q + 1]. Finally, let
F(n, g) = [C(n, g) - nC(q, g/2)]/2n,
where C(k, j) is the usual binomial coefficient k!/[j!(k-j)!]. The conjecture is
F(n, g) = 1 (mod q) if g = q + 1 and = 0 (mod q) if not.