To analyze questions like this
On variations of Erdős squarefree conjecture: presentation and a question as a simple case
it would be nice to have a simple criterion when $\binom{n}{m}$ is divisible by $q^2$, when $q$ is a prime and $n>m>1$ integers. We can assume $2m\le n$. Maybe creating pascal's triangle modulo $q^2$ gives a structure.
Does anyone know a criterion that is easy to check without calculating the binomial coefficient ?