I was looking at two sequences of integers, both with prominent place is combinatorics. The first one appears, for instance, in Stieltjes moment sequences for pattern-avoiding permutations (see page 23) $$a_n=\sum_{k=0}^n\frac{\binom{2k}k\binom{n+1}{k+1}\binom{n+2}{k+1}}{(n+1)^2(n+2)}.$$ The second appears in lattice path enumerations as Motzkin numbers (a close cousin of Catalan numbers) $$b_n=\sum_{k=0}^{\lfloor\frac{n}2\rfloor}\frac{\binom{n}{2k}\binom{2k}k}{k+1}.$$

In modulo $2$ arithmetic, I run into what seems to be a (happy) coincidence. Let me ask:

QUESTION.Is this true? $$a_n\equiv b_n \mod 2.$$

**ADDED.** $a_n$ or $b_n$ is even iff $n$ is part of this sequence listed on OEIS. In fact, whenever that happens, $\nu_2(a_n)=1$ and $\nu_2(b_n)\in\{1,2\}$.