Congruences involving quadrinomial coefficients

For nonnegative integers n and k, one defines the quadrinomial coefficient \(\left( {\begin{array}{c}n\\ k\end{array}}\right) _{3}\) ${\left(\begin{array}{c}n\\ k\end{array}\right)}_3$ as the coefficient of \(x^k\) $x^k$ in the polynomial expansion of \(\left( 1+x+x^2+x^3\right) ^{n}.\) ${\left(1,+,x,+,x^2,+,x^3\right)}^n.$ In this paper, we establish congruences (mod \(p^2\) $p^2$ ) involving the quadrinomial coefficients \(\genfrac(){0.0pt}0{np-1}{p-1}_{3}\) ${\left({\displaystyle \genfrac{}{}{0.0pt}{}{np-1}{p-1}}\right)}_3$ and  \(\genfrac(){0.0pt}0{np-1}{\frac{p-1}{2}}_{3}.\) ${\left({\displaystyle \genfrac{}{}{0.0pt}{}{np-1}{\frac{p-1}{2}}}\right)}_3.$ This extends some known congruences involving the binomial and trinomial coefficients.