Divisibility of some binomial sums

Abstract

With help of $q$-congruence, we prove the divisibility of some binomial sums. For example, for any integers $\rho ,n\ge 2$, $$\sum _{k=0}^{n-1}(4k+1){\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^{\rho }\cdot (-4{)}^{\rho (n-1-k)}\equiv 0(\mathrm{mod}{2}^{\rho -2}n\left(\genfrac{}{}{0ex}{}{2n}{n}\right)).$$

This is a very preliminary, which maybe contains some minor mistakes