Many famous sequences $\{a_n\}$ satisfy recurrence relations. For example, the Fibonacci numbers $\{0,1,1,2,3,5,\ldots\}$ and Lucas numbers $\{2,1,3,4,7,11,\ldots\}$ both satisfy $$ a_n = a_{n-1} + a_{n-2}, $$ the Pell numbers $\{0,1,2,5,12,29,\ldots\}$ satisfy $$ a_n = 2a_{n-1} + a_{n-2}, $$ the Tribonacci numbers $\{0,1,1,2,4,7,\ldots\}$ satisfy $$ a_n = a_{n-1} + a_{n-2} + a_{n-3}, $$ and the Padovan numbers $\{1,1,1,2,2,3,4,5,7,\ldots\}$ satisfy $$ a_n = a_{n-2} + a_{n-3}. $$ The characteristic equations of these four recurrences are, respectively, \begin{align*} x^2 & = x + 1, \\ x^2 & = 2x + 1, \\ x^3 & = x^2 + x + 1, \\ x^3 & = x + 1. \end{align*} All four of these characteristic equations have something in common. In each of these cases, precisely one of the roots has modulus greater than 1, and the rest have modulus less than 1. (This is related to the concept of a "Pisot number".) If we wanted to give a suggestive short name to this property, we could perhaps call it the "one big root" property. The "one big root" property makes it convenient to describe the asymptotic behavior of the sequence, because powers of the "small roots" approach 0.
I am seeking a reference for a sufficient condition on the coefficients of the polynomial that guarantee that it has the "one big root" property. I once read an article that proved such a condition, but now I cannot find it. The condition, paraphrased imprecisely using just my memory, was that the coefficients on the right side be positive and nonincreasing, like 1,1,1 or 2,1 but not 1,2. (I don't remember whether they considered the possibility of some coefficients on the right side being 0.)
If anyone can help me find the reference (which phrased their results in the context of asymptotics of recurrence relations), I would be greatly appreciative. My best memory is that it was from the current millennium, but there's a chance it was actually the late 20th century.