Define the sequence $b_1=1$ and $$b_n=\sum_{k=1}^{n-1}\binom{n-1}k\binom{n-1}{k-1}b_kb_{n-k}.$$

By now, there is enough in the literature that $C_n$ is odd iff $n=2^k-1$ for some $k$ where $C_n$ are the Catalan numbers: $C_0=1$ and $$C_{n+1}=\sum_{k=0}^nC_kC_{n-k}.$$

In the same spirit, I ask:

QUESTION. is it true that $b_n$ is odd iff $n=2^k$ for some $k$?