Elementary proof of congruences involving trinomial coefficients for babbage and morley

Abstract

The aim of this work is to establish congruences $(\mathrm{mod}{p}^2)$ involving the trinomial coefficients ${(\genfrac{}{}{0ex}{}{np-1}{p-1})}_2$ and ${(\genfrac{}{}{0ex}{}{np-1}{(p-1)/2})}_2$ arising from the expansion of the powers of the polynomial $1+x+x^2$. In main results, we extend some known congruences involving the binomial coefficients $(\genfrac{}{}{0ex}{}{np-1}{p-1})$ and $(\genfrac{}{}{0ex}{}{np-1}{(p-1)/2})$, and establish congruences linking binomial coefficients and harmonic numbers.