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][1] and further extended by [Postnikov][2]. Let's introduce the sequence $T_n=\frac2{n(n+1)}\binom{4n+1}{n-1}$ which count 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 $n=2^h-1$?




[1]:https://users.math.msu.edu/users/bsagan/Papers/Old/ccm.pdf
[2]:http://www-math.mit.edu/~apost/papers/weightedCatalan.pdf