Quadrinomial-Like Versions for Wolstenholme, Morley and Glaisher Congruences

The quadrinomial coefficient is defined as the coefficient of $x^k$ in the polynomial expansion of $\left( 1+x+x^2+x^3\right) ^{n}$, where $n$ and $k$ are nonnegative integers. In the present paper, we derive some congruences involving the quadrinomial coefficients.

For instance, we establish two congruences that are analogous to those of Morley and Wolstenholme.