A generic approach to proving Tur\'{a}n-type inequalities for sequences that admit exact formulas, with an application to unimodal sequences

We derive an asymptotic expansion with effective error bound for $u(n)$, counting the number of unimodal sequences of size $n$. We prove that $u(n)$ satisfies the higher order Tur\'{a}n inequalities for $n\geq33$ and that certain second $j$-shifted difference of $u(n)$ are positive.