  We show that the number $A(n,m)$ of partitions with $m$ even parts and
largest hook length $n$ is strongly unimodal with mode [(n-1)/4] for $n\ge 6$.
We establish this result by induction, using a $5$-term recurrence due to Lin,
Xiong and Yan, and two $4$-term recurrences obtained by Zeilberger's algorithm.
The sequence $A(n,m)$ is not log-concave. Using M\"obius transformation and the
method of interlacing zeros, we obtain that every zero of every generating
function $\sum_m A(n,m)z^m$ lies on the left half part of the circle |z-1|=2.
Moreover, as a direct application of Wang and Zhang's characterization of root
geometry of polynomial sequences that satisfy a recurrence of type $(1,1)$, we
see that all these zeros are densely distributed on the half circle.
