Some congruences involving binomial coefficients

Abstract

Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let $p>3$ be a prime. We show that $$T_{p-1}\equiv \left(\frac{p}{3}\right)3^{p-1}(\mathrm{mod}{p}^2),$$ 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 $$\sum _{k=0}^{p-1}\left(\genfrac{}{}{0ex}{}{p-1}{k}\right)\left(\genfrac{}{}{0ex}{}{2k}{k}\right)((-1{)}^k-(-3{)}^{-k})\equiv \left(\frac{p}{3}\right)(3^{p-1}-1)(\mathrm{mod}{p}^3).$$ In addition, we get some new combinatorial identities.

9 pages, final published version

References in corpus (1)

• On sums of binomial coefficients modulo p^2

Cited by in corpus (1)

• Open Conjectures on Congruences