Supercongruences involving dual sequences

In this paper we study some sophisticated supercongruences involving dual sequences. For $n=0,1,2,\dots$ define $$d_n(x)=\sum _{k=0}^n\left(\genfrac{}{}{0px}{}{n}{k}\right)\left(\genfrac{}{}{0px}{}{x}{k}\right)2^k$$ and $$s_n(x)=\sum _{k=0}^n\left(\genfrac{}{}{0px}{}{n}{k}\right)\left(\genfrac{}{}{0px}{}{x}{k}\right)\left(\genfrac{}{}{0px}{}{x+k}{k}\right)=\sum _{k=0}^n\left(\genfrac{}{}{0px}{}{n}{k}\right)(-1{)}^k\left(\genfrac{}{}{0px}{}{x}{k}\right)\left(\genfrac{}{}{0px}{}{-1-x}{k}\right)\mathrm{.}$$ For any odd prime $p$ and $p$-adic integer $x$, we determine ${\sum }_{k=0}^{p-1}(\pm 1{)}^k{d}_k(x{)}^2$ and ${\sum }_{k=0}^{p-1}(2k+1)d_k(x{)}^2$ modulo $p^2$; for example, we establish the new $p$-adic congruence $$\sum _{k=0}^{p-1}(-1{)}^k{d}_k(x{)}^2\equiv (-1{)}^{⟨x{⟩}_p}(\mathrm{m}\mathrm{o}\mathrm{d}{p}^2),$$ where $⟨x{⟩}_p$ denotes the least nonnegative integer $r$ with $x\equiv r(\mathrm{m}\mathrm{o}\mathrm{d}p)$. For any prime $p>3$ and $p$-adic integer $x$, we determine ${\sum }_{k=0}^{p-1}{s}_k(x{)}^2$ modulo $p^2$ (or $p^3$ if $x\in \{0,\dots ,p-1\}$), and show that $$\sum _{k=0}^{p-1}(2k+1)s_k(x{)}^2\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}{p}^2)\mathrm{.}$$ We also pose several related conjectures.