Some congruences involving binomial coefficients
Volume 139 / 2015
Colloquium Mathematicum 139 (2015), 127-136
MSC: Primary 11A07, 11B65; Secondary 05A10, 05A19, 11B39.
DOI: 10.4064/cm139-1-8
Abstract
Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let $p>3$ be a prime. We show that $$T_{p-1}\equiv\bigg(\frac p3\bigg)3^{p-1}\pmod{p^2},$$ 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 $$\sum_{k=0}^{p-1}\binom{p-1}k\binom{2k}k((-1)^k-(-3)^{-k})\equiv \bigg(\frac p3\bigg)(3^{p-1}-1)\pmod{p^3}.$$ In addition, we get some new combinatorial identities.
