Consider the ubiquitous Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$. In this post, I am looking for your help in my attempt to collect alternative proofs of the following fact: *$C_n$ is odd if and only if $n=2^r-1$.*

Proofs that I know:

1. an application of Legendre's (Kummer) formula $\nu_2(C_n)=s(n+1)-1$, where $s(n)$ is the sum of the binary digits of $n$.

2. See E. Deutsch and B. Sagan, [Congruences for Catalan and Motzkin numbers and related sequences][1].

3. See A. Postnikov and B. Sagan, [What power of two divides a weighted Catalan number?][2]

4. See O. Egecioglu, [The parity of the Catalan number via lattice paths.][3] 

>**QUESTION.** Do you know of or can provide a new proof that $C_n$ is odd if and only if $n=2^r-1$?

[1]: https://arxiv.org/pdf/math/0407326.pdf
[2]: https://arxiv.org/pdf/math/0601339.pdf
[3]: https://www.fq.math.ca/Scanned/21-1/egecioglu.pdf