Divisibility of some binomial sums

Abstract

With the help of $q$-congruences, we consider divisibility of some binomial sums. For example, for any integers $\rho \geq 2$ and $n\geq 2$, \begin{equation} \sum _{k=0}^{n-1}(4k+1)\binom {2k}{k}^\rho \cdot (-4)^{\rho (n-1-k)}\equiv 0\ \biggl (\!{\rm mod}

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