[math-fun] Question about Motzkin nmubers from Vladimir Retakh

Dear Math Fun, My Rutgers colleague Vladimir Retakh ( vretakh@math.rutgers.edu) has a question about Motzkin numbers; He asks: I consider Dyck paths on the integer grid from (0,0) to (n,n) consisting of zigzags with horizontal and vertical segments and staying below the diagonal x=y. I claim that the number of such paths with horizontal segments of length non-equal to 3, 5, 7, ... (odd numbers greater than 1) is equal to the n-th Motzkin numbers. May I have a reference confirming this statement? Me: The literature about these numbers is huge (see https://oeis.org/A001006, which is about the second-biggest entry after the Catalan numbers). Can someone help Professor Retakh with a reference or proof?