# Asymptotic densities of Catalan numbers modulo 2^k

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

And so

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.

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