  We compute the number of rhombus tilings of a hexagon with side lengths
N,M,N,N,M,N, with N and M having the same parity, which contain a particular
rhombus next to the center of the hexagon. The special case N=M of one of our
results solves a problem posed by Propp. In the proofs, Hankel determinants
featuring Bernoulli numbers play an important role.
