Congruences concerning quadrinomial coefficients

Abstract

In this paper, we establish congruences (mod $p^2$) involving the quadrinomial coefficients $\dbinom{np-1}{p-1}_{3}$ and $\dbinom{np-1}{\frac{p-1}{2}}_{3}$. This is an analogue of congruences involving the trinomial coefficients $\dbinom{np-1}{p-1}_{2}$ and $\dbinom{np-1}{\frac{p-1}{2}}_{2}$ due to Elkhiri and Mihoubi.