The Integer Sequence Transform $a \mapsto b$ where $b_n$ is the Number of Real Roots of the Polynomial $a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$

Abstract

We discuss the integer sequence transform $a \mapsto b$ where $b_n$ is the number of real roots of the polynomial $a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$. It is shown that several sequences $a$ give the trivial sequence $b = (0,1,0,1, 0,1,\ldots)$, i.e., ${b_n = n \bmod 2}$, among them the Catalan numbers, central binomial coefficients, $n!$ and $\binom{n+k}{n}$ 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 $k \geq 3$. Further, general procedures are given for constructing real sequences $a_n$ for which $b_n$ is either always maximal or minimal.