Elementary proof of congruences involving trinomial coefficients for babbage and morley

The aim of this work is to establish congruences \( \pmod{p^2} \) involving the trinomial coefficients \( \binom{np-1}{p-1}_2 \) and \( \binom{np-1}{(p-1)/2}_2 \) 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 \( \binom{np-1}{p-1} \) and \( \binom{np-1}{(p-1)/2} \), and establish congruences linking binomial coefficients and harmonic numbers.