A Stern-type congruence for the Schröder numbers

The Schröder number Sn counts the number of the lattice paths from (0,0) to (n,n), containing no point above the line y=x and using only steps (1,0), (0,1) and (1,1). We prove that Sn+2α≡Sn+2α+1(mod2α+2),where n≥1 and α≥1.