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 in Congruences for Catalan and Motzkin numbers and related sequences and further extended by Postnikov and Sagan in What power of two divides a weighted Catalan number?.
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 (see Bousquet-Mélou, Fusy, and Ratelle - The number of intervals in the m-Tamari lattices) (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 $n=2^h-1$?
