A unimodal sequence with mode at a quarter length

We show that the number (Formula presented.) of partitions with m even parts and largest hook length n is strongly unimodal with mode (Formula presented.) for (Formula presented.). 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 (Formula presented.) is not log-concave. Using Möbius transformation and the method of interlacing zeros, we obtain that every zero of every generating polynomial (Formula presented.) lies on the left half part of the circle (Formula presented.). Moreover, as an application of Wang and Zhang's characterization of root geometry of polynomial sequences that satisfy a recurrence, we confirm that the zeros are densely distributed on the half circle.