Two new kinds of numbers and related divisibility results

We mainly introduce two new kinds of numbers given by $$R_n=\sum _{k=0}^n\left(\genfrac{}{}{0px}{}{n}{k}\right)\left(\genfrac{}{}{0px}{}{n+k}{k}\right)\frac{1}{2k-1}(n=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.})$$ and $$S_n=\sum _{k=0}^n{\left(\genfrac{}{}{0px}{}{n}{k}\right)}^2\left(\genfrac{}{}{0px}{}{2k}{k}\right)(2k+1)(n=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.})\mathrm{.}$$ We find that such numbers have many interesting arithmetic properties. For example, if $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ is a prime with $p=x^2+y^2$ (where $x\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and $y\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}2)$), then $$R_{(p-1)\mathrm{/}2}\equiv p-(-1{)}^{(p-1)\mathrm{/}4}2x(\mathrm{m}\mathrm{o}\mathrm{d}{p}^2)\mathrm{.}$$ Also, $$\frac{1}{n^2}\sum _{k=0}^{n-1}{S}_k\in \mathbb{Z}and\frac{1}{n}\sum _{k=0}^{n-1}{S}_k(x)\in \mathbb{Z}[x]\text{for all }n=1,2,3,\mathrm{.}\mathrm{.}\mathrm{.},$$ where $S_k(x)={\sum }_{j=0}^k{\left(\genfrac{}{}{0px}{}{k}{j}\right)}^2\left(\genfrac{}{}{0px}{}{2j}{j}\right)(2j+1)x^j$. For any positive integers $a$ and $n$, we show that, somewhat surprisingly, $$\frac{1}{n^2}\sum _{k=0}^{n-1}(2k+1){\left(\genfrac{}{}{0px}{}{n-1}{k}\right)}^a{\left(\genfrac{}{}{0px}{}{-n-1}{k}\right)}^a\in \mathbb{Z}and\frac{1}{n}\sum _{k=0}^{n-1}\frac{{\left(\genfrac{}{}{0px}{}{n-1}{k}\right)}^a{\left(\genfrac{}{}{0px}{}{-n-1}{k}\right)}^a}{4k^2-1}\in \mathbb{Z}\mathrm{.}$$ We also solve a conjecture of V.J.W. Guo and J. Zeng, and pose several conjectures for further research.