Given a sequence of integers one gives a Dyck path P of length 2n the weight where is the height of the ith ascent of P. The corresponding weighted Catalan number is where the sum is over all Dyck paths of length 2n. So, in particular, the ordinary Catalan numbers correspond to for all . Let stand for the base two exponent of n, i.e., the largest power of 2 dividing n. We give a condition on b which implies that . In the special case , this settles a conjecture of Postnikov about the number of plane Morse links. Our proof generalizes the recent combinatorial proof of Deutsch and Sagan of the classical formula for .