On balanced colorings of the n-cube

A 2-coloring of the n-cube in the n-dimensional Euclidean space can be considered as an assignment of weights of 1 or 0 to the vertices. Such a colored ncube is said to be balanced if its center of mass coincides with its geometric center. Let Bn,2k be the number of balanced 2-colorings of the n-cube with 2k vertices having weight 1. Palmer, Read, and Robinson conjectured that for n ≥ 1, the sequence {Bn,2k}k=0,1,...,2n−1 is symmetric and unimodal. We give a proof of this conjecture. We also propose a conjecture on the log-concavity of Bn,2k for fixed k, and by probabilistic method we show that it holds when n is sufficiently large.