Congruences for the second-order Catalan numbers

Authors:
Li-Lu Zhao, Hao Pan and Zhi-Wei Sun

Journal:
Proc. Amer. Math. Soc. **138** (2010), 37-46

MSC (2000):
Primary 11B65; Secondary 05A10, 11A07

Published electronically:
September 4, 2009

MathSciNet review:
2550168

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be any odd prime. We mainly show that

**[Gr]**Andrew Granville,*Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers*, Organic mathematics (Burnaby, BC, 1995) CMS Conf. Proc., vol. 20, Amer. Math. Soc., Providence, RI, 1997, pp. 253–276. MR**1483922****[HT]**Charles Helou and Guy Terjanian,*On Wolstenholme’s theorem and its converse*, J. Number Theory**128**(2008), no. 3, 475–499. MR**2389852**, 10.1016/j.jnt.2007.06.008**[HS]**Hong Hu and Zhi-Wei Sun,*An extension of Lucas’ theorem*, Proc. Amer. Math. Soc.**129**(2001), no. 12, 3471–3478 (electronic). MR**1860478**, 10.1090/S0002-9939-01-06234-7**[PS]**Hao Pan and Zhi-Wei Sun,*A combinatorial identity with application to Catalan numbers*, Discrete Math.**306**(2006), no. 16, 1921–1940. MR**2251572**, 10.1016/j.disc.2006.03.050**[St]**Richard P. Stanley,*Enumerative combinatorics. Vol. 2*, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR**1676282****[ST1]**Z.-W. Sun and R. Tauraso,*On some new congruences for binomial coefficients*, Acta Arith., to appear.`http://arxiv.org/abs/0709.1665`.**[ST2]**Z.-W. Sun and R. Tauraso,*New congruences for central binomial coefficients*, preprint,`http://arxiv.org/abs/0805.0563`.

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Additional Information

**Li-Lu Zhao**

Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China

Email:
zhaolilu@gmail.com

**Hao Pan**

Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China

Email:
haopan79@yahoo.com.cn

**Zhi-Wei Sun**

Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China

Email:
zwsun@nju.edu.cn

DOI:
http://dx.doi.org/10.1090/S0002-9939-09-10067-9

Received by editor(s):
April 7, 2009

Published electronically:
September 4, 2009

Additional Notes:
The third author is the corresponding author. He was supported by the National Natural Science Foundation (grant 10871087) and the Overseas Cooperation Fund of China.

Communicated by:
Ken Ono

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.