  The numbers $R_n$ and $W_n$ are defined as \begin{align*}
R_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{1}{2k-1},\ \text{and}\
W_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{3}{2k-3}. \end{align*} We
prove that, for any positive integer $n$ and odd prime $p$, there hold
\begin{align*} \sum_{k=0}^{n-1}(2k+1)R_k^2 &\equiv 0 \pmod{n}, \\
\sum_{k=0}^{p-1}(2k+1)R_k^2 &\equiv 4p(-1)^{\frac{p-1}{2}} -p^2 \pmod{p^3}, \\
9\sum_{k=0}^{n-1}(2k+1)W_k^2 &\equiv 0 \pmod{n}, \\ \sum_{k=0}^{p-1}(2k+1)W_k^2
&\equiv 12p(-1)^{\frac{p-1}{2}}-17p^2 \pmod{p^3}, \quad\text{if $p>3$.}
\end{align*} The first two congruences were originally conjectured by Z.-W.
Sun. Our proof is based on the multi-variable Zeilberger algorithm and the
following observation: $$ {2n\choose n}{n\choose k}{m\choose k}{k\choose
m-n}\equiv 0\pmod{{2k\choose k}{2m-2k\choose m-k}}, $$ where $0\leqslant
k\leqslant n\leqslant m \leqslant 2n$.
