The Catalan numbers are defined by
Corollary 2 of the previous post established that
This result can be used to show that the asymptotic density of the set is
To this end let for some Then and
So, as mentioned in the previous post
where is a polynomial in of degree with coefficients depending on . Since is fixed and for fixed the second term above is zero in the limit and
This result is not surprising given the highly composite nature of the Catalan numbers. In general we would also expect that the same result holds with replaced by any natural number.