Some divisibility properties of binomial and q-binomial coefficients

We first prove that if a has a prime factor not dividing b then there are infinitely many positive integers n   such that (an+bnan) is not divisible by bn+1bn+1. This confirms a recent conjecture of Z.-W. Sun. Moreover, we provide some new divisibility properties of binomial coefficients: for example, we prove that (12n3n) and (12n4n) are divisible by 6n−16n−1, and that (330n88n) is divisible by 66n−166n−1, for all positive integers n. As we show, the latter results are in fact consequences of divisibility and positivity results for quotients of q-binomial coefficients by q-integers, generalising the positivity of q-Catalan numbers. We also put forward several related conjectures.