Some congruences involving binomial coefficients

Hui-Qin Cao, and Zhi-Wei Sun. "Some congruences involving binomial coefficients." Colloquium Mathematicae 139.1 (2015): 127-136. <http://eudml.org/doc/284057>.

@article{Hui2015,
abstract = {Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that $T_\{p-1\} ≡ (p/3) 3^\{p-1\} (mod p²)$, where the central trinomial coefficient Tₙ is the constant term in the expansion of $(1 + x + x^\{-1\})ⁿ$. We also prove three congruences modulo p³ conjectured by Sun, one of which is $∑_\{k=0\}^\{p-1\} \binom\{p-1\}\{k\}\binom\{2k\}\{k\} ((-1)^k - (-3)^\{-k\}) ≡ (p/3)(3^\{p-1\} - 1) (mod p³)$. In addition, we get some new combinatorial identities.},
author = {Hui-Qin Cao, Zhi-Wei Sun},
journal = {Colloquium Mathematicae},
keywords = {congruences; binomial coefficients; Lucas sequences; central trinomial coefficients},
language = {eng},
number = {1},
pages = {127-136},
title = {Some congruences involving binomial coefficients},
url = {http://eudml.org/doc/284057},
volume = {139},
year = {2015},
}

TY - JOUR
AU - Hui-Qin Cao
AU - Zhi-Wei Sun
TI - Some congruences involving binomial coefficients
JO - Colloquium Mathematicae
PY - 2015
VL - 139
IS - 1
SP - 127
EP - 136
AB - Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that $T_{p-1} ≡ (p/3) 3^{p-1} (mod p²)$, where the central trinomial coefficient Tₙ is the constant term in the expansion of $(1 + x + x^{-1})ⁿ$. We also prove three congruences modulo p³ conjectured by Sun, one of which is $∑_{k=0}^{p-1} \binom{p-1}{k}\binom{2k}{k} ((-1)^k - (-3)^{-k}) ≡ (p/3)(3^{p-1} - 1) (mod p³)$. In addition, we get some new combinatorial identities.
LA - eng
KW - congruences; binomial coefficients; Lucas sequences; central trinomial coefficients
UR - http://eudml.org/doc/284057
ER -