A Stern-type congruence for the Schroder numbers

For the Schr\"oder number $$S_n=\sum_{k=0}^n\binom{n}k\binom{n+k}k\frac1{k+1},$$ we prove that $$S_{n+2^\alpha}\equiv S_{n}+2^{\alpha+1}\pmod{2^{\alpha+2}},$$ where $n\geq 1$ and $\alpha\geq 1$.