Let $C_n=\frac1{n+1}\binom{2n}n$ be the Catalan numbers. Recall, in particular, that $C_n$ is odd iff $n=2^h-1$. A combinatorial proof is given by Deutsch and Sagan and further extended by Postnikov and Sagan.

Let's introduce the sequence $T_n=\frac2{n(n+1)}\binom{4n+1}{n-1}$ which enumerates intervals of the so-called Tamari lattice (also counting triangular maps).

It is rather simple to prove the following using basic arithmetic means.

QUESTION. Can you provide a combinatorial justification that $T_n$ is odd if $n=2^h-1$?