Some congruences involving binomial coefficients

TL;DR

This paper investigates congruences involving binomial and trinomial coefficients modulo prime powers, proving new results and conjectures, and deriving novel combinatorial identities.

Contribution

It establishes new congruences for central trinomial coefficients modulo p^2 and p^3, confirming conjectures by Sun and introducing new combinatorial identities.

Findings

Proved that T_{p-1} ≡ (p/3) * 3^{p-1} mod p^2.

Confirmed three conjectured congruences modulo p^3 by Sun.

Derived new combinatorial identities related to binomial coefficients.

Abstract

Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let $p>3$ be a prime. We show that $$T_{p-1}\equiv\left(\frac p3\right)3^{p-1}\ \pmod{p^2},$$ where the central trinomial coefficient $T_n$ is the constant term in the expansion of $(1+x+x^{-1})^n$. We also prove three congruences modulo $p^3$ conjectured by Sun, one of which is $$\sum_{k=0}^{p-1}\binom{p-1}k\binom{2k}k((-1)^k-(-3)^{-k})\equiv \left(\frac p3\right)(3^{p-1}-1)\ \pmod{p^3}.$$ In addition, we get some new combinatorial identities.