A Stern-type congruence for the Schroder numbers

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

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