A combinatorial identity with application to Catalan numbers

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By a very simple argument, we prove that if l, m, n {0, 1, 2, . . . } then l k=0 (-1)m-k l k m - k n 2k k - 2l + m = l k=0 l k 2k n n - l m + n - 3k - l . On the basis of this identity, for d, r {0, 1, 2, . . . } we construct explicit F(d, r) and G(d, r) such that for any prime p > max{d, r} we have p-1 k=1 kr Ck+d F(d, r) (mod p) if p 1 (mod 3), G(d, r) (mod p) if p 2 (mod 3), where Cn denotes the Catalan number 1 n+1 2n n . For example, when p 5 is a prime, we have p-1 k=1 k2 Ck -2/3 (mod p) if p 1 (mod 3), -1/3 (mod p) if p 2 (mod 3); and 0<k<p-4 Ck+4 k 503/30 (mod p) if p 1 (mod 3), -100/3 (mod p) if p 2 (mod 3). This paper also contains some new recurrence relations for Catalan numbers. Key words and phrases. Binomial coefficient; Combinatorial identity; Catalan number. 2000 Mathematics Subject Classification. Primary 11B65; Secondary 05A10, 05A19, 11A07, 11B37. The second author is responsible for communications, and partially supported by the National Science F...

Hao Pan, Zhi-Wei Sun

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