The Integer Sequence Transform $a\mapsto b$ where $b_n$ is the Number of Real Roots of the Polynomial $a_0+a_{1}x+a_2{x}^2+\cdots +a_n{x}^n$

Abstract

We discuss the integer sequence transform $a\mapsto b$ where $b_n$ is the number of real roots of the polynomial $a_0+a_{1}x+a_2{x}^2+\cdots +a_n{x}^n$. It is shown that several sequences $a$ give the trivial sequence $b=(0,1,0,1,0,1,\dots )$, i.e., $b_n=n\mathrm{mod}2$, among them the Catalan numbers, central binomial coefficients, $n!$ and $\left(\genfrac{}{}{0ex}{}{n+k}{n}\right)$ for a fixed $k$. We also look at some sequences $a$ for which $b$ is more interesting such as $a_n=(n+1{)}^k$ for $k\ge 3$. Further, general procedures are given for constructing real sequences $a_n$ for which $b_n$ is either always maximal or minimal.