Binary Partitions Revisited

The restricted binary partition function bk(n) enumerates the number of ways to represent n as n=2a0+2a1+…+2aj with 0⩽a0⩽a1⩽…⩽aj<k. We study the question of how large a power of 2 divides the difference bk(2r+2n)−bk−2(2rn) for fixed k⩾3, r⩾1, and all n⩾1.