Congruences for the second-order Catalan numbers
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- by Li-Lu Zhao, Hao Pan and Zhi-Wei Sun PDF
- Proc. Amer. Math. Soc. 138 (2010), 37-46 Request permission
Abstract:
Let $p$ be any odd prime. We mainly show that \begin{equation*} \sum _{k=1}^{p-1}\frac {2^{k}}k\binom {3k}k\equiv 0\ (\operatorname {mod} p) \end{equation*} and \begin{equation*} \sum _{k=1}^{p-1}2^{k-1} C_{k}^{(2)}\equiv (-1)^{(p-1)/2}-1\ (\operatorname {mod} p),\end{equation*} where $C_{k}^{(2)}=\binom {3k}k/(2k+1)$ is the $k$th Catalan number of order 2.References
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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
- MR Author ID: 254588
- Email: zwsun@nju.edu.cn
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 37-46
- MSC (2000): Primary 11B65; Secondary 05A10, 11A07
- DOI: https://doi.org/10.1090/S0002-9939-09-10067-9
- MathSciNet review: 2550168