منابع مشابه
Some Congruences Involving Binomial Coefficients
Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that Tp−1 ≡ (p 3 ) 3p−1 (mod p), 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 p conjectured by Sun, one of which is p−1 ∑ k=0 ( p− 1 k )( 2k k ) ((−1) − (−3)−k) ≡ (p 3 ) (3p−1 −...
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In this paper we establish some new congruences involving central binomial coefficients as well as Catalan numbers. Let p be a prime and let a be any positive integer. We determine ∑pa−1 k=0 ( 2k k+d ) mod p2 for d = 0, . . . , pa and ∑pa−1 k=0 ( 2k k+δ ) mod p3 for δ = 0, 1. We also show that
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Let p > 3 be a prime. We show that T p−1 ≡ p 3 3 p−1 (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 conjectured by Sun one of which is as follows: p−1 k=0 p − 1 k 2k k ((−1) k − (−3) −k) ≡ p 3 (3 p−1 − 1) (mod p 3).
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Motivated by recent works of Sun and Tauraso, we prove some variations on the Green-Krammer identity involving central q-binomial coefficients, such as n−1 ∑ k=0 (−1)kq−(k+1 2 ) [ 2k k ] q ≡ (n 5 ) q−bn 4/5c (mod Φn(q)), where ( n p ) is the Legendre symbol and Φn(q) is the nth cyclotomic polynomial. As consequences, we deduce that 3am−1 ∑
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ژورنال
عنوان ژورنال: Applied Mathematics Letters
سال: 1991
ISSN: 0893-9659
DOI: 10.1016/0893-9659(91)90043-u