The Motzkin numbers are defined by
where are the Catalan numbers. The following result is established in 
Theorem 1 (Theorem 5.5 of ). The nth Motzkin number is even if and only if for and . Moreover, we have
where is the number of digit 1’s in the base 2 representation of .
Remark 2. The 4 choices for in the above theorem give 4 disjoint sets of numbers .
Theorem 3. Each of the 4 disjoint sets defined by the choice of in Theorem 1 has asymptotic density in the natural numbers.
Proof: Use the result for the set of Theorem 1 of the post here with .
Corollary 4: The asymptotic density of
The asymptotic density of
The asymptotic density of each the sets
and is .
Remark 5: The first 2 statements of Corollary 4 follows immediately from theorems 1 and 3. The third statement follows from the observation that the numbers of 1’s in the base 2 expansion of is equally likely to be odd or even and therefore the same applies the the number of 1’s in the base 2 expansion of . Since the asymptotic density of the 2 sets combined is (from Theorems 1 and 3), each of the two sets has asymptotic density of .
 Eu, S.-P., Liu, S.-C., & Yeh, Y.-N. (2008). Catalan and Motzkin numbers modulo 4 and 8. European Journal of Combinatorics, 29(6), 1449–1466. http://doi.org/10.1016/j.ejc.2007.06.019