On sums of binomial coefficients involving Catalan and Delannoy numbers modulo $p^2$

Abstract

In this paper, we prove some congruences conjectured by Z.-W. Sun: For any prime $p>3$, we determine

$$\begin{aligned} \sum \limits _{k = 0}^{p - 1} {\frac{{{C_k}C_k^{(2)}}}{{{{27}^k}}}} \quad {\text { and }}\quad \sum \limits _{k = 1}^{p - 1} {\frac{{\left( {\begin{array}{l} {2k} \\ {k - 1} \\ \end{array}} \right) \left( { \begin{array}{l} {3k} \\ {k - 1} \\ \end{array} } \right) }}{{{{27}^k}}}} \end{aligned}$$

modulo $p^2$, where $C_k=\frac{1}{k+1}\left( {\begin{array}{c}2k\\ k\end{array}}\right)$ is the k-th Catalan number and $C_k^{(2)}=\frac{1}{2k+1}\left( {\begin{array}{c}3k\\ k\end{array}}\right)$ is the second-order Catalan numbers of the first kind. And we prove that

$$\begin{aligned} \sum _{k=1}^{p-1}\frac{D_k}{k}\equiv -q_p(2)+pq_p(2)^2\pmod {p^2}, \end{aligned}$$

where $D_n=\sum _{k=0}^{n}\left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}n+k\\ k\end{array}}\right)$ is the n-th Delannoy number and $q_p(2)=(2^{{p-1}}-1)/p$ is the Fermat quotient.