Some congruences involving binomial coefficients

Hui-Qin Cao¹ ; Zhi-Wei Sun²

¹ Department of Applied Mathematics Nanjing Audit University Nanjing 211815 People's Republic of China
² Department of Mathematics Nanjing University Nanjing 210093 People's Republic of China

Colloquium Mathematicum, Tome 139 (2015) no. 1, pp. 127-136

Résumé

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.

Détail

DOI : 10.4064/cm139-1-8

Keywords: binomial coefficients central trinomial coefficients play important roles combinatorics prime p equiv bigg frac bigg p pmod where central trinomial coefficient constant term expansion prove three congruences modulo conjectured sun which sum p binom p binom k k equiv bigg frac bigg p pmod addition get combinatorial identities