Consider the ubiquitous Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$. In this post, I am very interesting and looking for your help in my attempt to collect alternative proofs of the following fact: $C_n$ is odd if an 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 $s(n)=$number of $1$'s in the binary expansion of $n$.

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

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

4. See O. Egecioglu, The parity of the catalan number via lattice paths.

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