A Stern-type congruence for the Schroder numbers

For the Schr"oder number $$S_n=\sum _{k=0}^n\left(\genfrac{}{}{0px}{}{n}{k}\right)\left(\genfrac{}{}{0px}{}{n+k}{k}\right)\frac{1}{k+1},$$ we prove that $$S_{n+2^{\alpha }}\equiv S_n+2^{\alpha +1}(\mathrm{m}\mathrm{o}\mathrm{d}{2}^{\alpha +2}),$$ where $n\ge 1$ and $\alpha \ge 1$.