TITLE: Logconcavity of height of Dyck paths
QUESTION [12 upvotes]: A finite sequence $a_i$ is called logconvace in case $a_i^2 \geq a_{i-1} a_{i+1}$.

Question : For a fixed $n$, is the sequence $a_{n,k}$ giving the number of Dyck paths of semilength $n$ having height $k$ logconcave? (see http://oeis.org/A080936)

REPLY [2 votes]: A stronger property than log-concavity, is real-rootedness of $\sum_k t^k a_{n,k}$.
However, for $n=4$, this polynomial is $1 + 7 t + 5 t^2 + t^3$ which is not real-rooted.