Congruences involving Delannoy numbers and Schröder numbers

The central Delannoy numbers $D_n={\sum }_{k=0}^n\left(\genfrac{}{}{0px}{}{n}{k}\right)\left(\genfrac{}{}{0px}{}{n+k}{k}\right)$ and the little Schröder number $s_n={\sum }_{k=1}^n\frac{1}{n}\left(\genfrac{}{}{0px}{}{n}{k}\right)\left(\genfrac{}{}{0px}{}{n}{k-1}\right)2^{n-k}$ are important quantities. In this paper, we confirm $$\frac{2}{3n(n+1)}\sum _{k=1}^n(-1{)}^{n-k}{k}^2{D}_k{D}_{k-1}\text{ and }\frac{1}{n}\sum _{k=1}^n(-1{)}^{n-k}(4k^2+2k-1)D_{k-1}{s}_k$$are positive odd integers for all $n=1,2,3,\cdots$. We also show that for any prime number $p>3$, $$\sum _{k=1}^{p-1}(-1{)}^k{k}^2{D}_k{D}_{k-1}\equiv -\frac{5}{6}p(\mathrm{m}\mathrm{o}\mathrm{d}{p}^2)$$ and $$\sum _{k=1}^p(-1{)}^k(4k^2+2k-1)D_{k-1}{s}_k\equiv -4p(\mathrm{m}\mathrm{o}\mathrm{d}{p}^2)\text{.}$$ Moreover, define \begin{equation*} s_n(x)=\sum_{k=1}^{n}\frac{1}{n}\binom{n}{k}\binom{n}{k-1}x^{k-1}(x+1)^{n-k}, \end{equation*} for any $n\in {\mathbb{Z}}^{+}$ is even we have \begin{equation*} \frac{4}{n(n+1)(n+2)(1+2x)^3}\sum_{k=1}^{n}k(k+1)(k+2)s_k(x)s_{k+1}(x)\in$\mathbb{Z}$[x]. \end{equation*}