Binomial coefficients, Catalan numbers and Lucas quotients
نویسندگان
چکیده
منابع مشابه
Binomial Coefficients , Catalan Numbers and Lucas Quotients
Let p be an odd prime and let a, m ∈ Z with a > 0 and p ∤ m. In this paper we determine p a −1 k=0 2k k+d /m k mod p 2 for d = 0, 1; for example, p a −1 k=0 2k k m k ≡ m 2 − 4m p a + m 2 − 4m p a−1 u p−(m 2 −4m p) (mod p 2), where (−) is the Jacobi symbol and {u n } n0 is the Lucas sequence given by u 0 = 0, u 1 = 1 and u n+1 = (m − 2)u n − u n−1 (n = 1, 2, 3,. . .). As an application, we deter...
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The binomial coefficients and Catalan triangle numbers appear as weight multiplicities of the finite-dimensional simple Lie algebras and affine Kac–Moody algebras. We prove that any binomial coefficient can be written as weighted sums along rows of the Catalan triangle. The coefficients in the sums form a triangular array, which we call the alternating Jacobsthal triangle. We study various subs...
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In this paper we consider the generalized Catalan numbers F (s, n) = 1 (s−1)n+1 ( sn n ) , which we call s-Catalan numbers. We find all natural numbers n such that for p prime, p divides F (p, n), q ≥ 1 and all distinct residues of F (p, n) (mod p), q = 1, 2. As a byproduct we settle a question of Hough and the late Simion on the divisibility of the 4-Catalan numbers by 4. We also prove that ( ...
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In this paper we consider the generalized Catalan numbers F (s, n) = 1 (s−1)n+1 ( sn n ) , which we call s-Catalan numbers. For p prime, we find all positive integers n such that p divides F (p, n), and also determine all distinct residues of F (p, n) (mod p), q ≥ 1. As a byproduct we settle a question of Hough and the late Simion on the divisibility of the 4-Catalan numbers by 4. In the second...
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ژورنال
عنوان ژورنال: Science China Mathematics
سال: 2010
ISSN: 1674-7283,1869-1862
DOI: 10.1007/s11425-010-3151-3