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

TL;DR

This paper develops a general method to prove Turán-type inequalities for sequences with explicit formulas, applying it to unimodal sequences and establishing inequalities for large n.

Contribution

It introduces a generic approach for proving Turán inequalities for sequences with exact formulas, demonstrated on unimodal sequences.

Findings

Proves that the number of unimodal sequences satisfies higher order Turán inequalities for n ≥ 33.

Establishes positivity of certain second j-shifted differences of u(n).

Provides an asymptotic expansion with error bounds for u(n).

Abstract

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.