Congruences for the second-order Catalan numbers
Congruences for the second-order Catalan numbers
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.