top
Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that
,
where the central trinomial coefficient Tₙ is the constant term in the expansion of . We also prove three congruences modulo p³ conjectured by Sun, one of which is
.
In addition, we get some new combinatorial identities.
@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 -