Congruences involving quadrinomial coefficients

For nonnegative integers n and k, one defines the quadrinomial coefficient \(\left( {\begin{array}{c}n\\ k\end{array}}\right) _{3}\) as the coefficient of \(x^k\) in the polynomial expansion of \(\left( 1+x+x^2+x^3\right) ^{n}.\) In this paper, we establish congruences (mod \(p^2\)) involving the quadrinomial coefficients \(\genfrac(){0.0pt}0{np-1}{p-1}_{3}\) and \(\genfrac(){0.0pt}0{np-1}{\frac{p-1}{2}}_{3}.\) This extends some known congruences involving the binomial and trinomial coefficients.