A Stern-type congruence for the Schroder numbers

Abstract

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 $α\geq 1$.