The Integer Sequence Transform a ↦ b where b_n is the Number of Real Roots of the Polynomial a₀ + a₁x + a₂x² + ⋯ + a_nxⁿ

2021/07/07 by W. Edwin Clark, Clark, W. Edwin, Mark Shattuck +1
Computer Science · Mathematics · Physics and Astronomy · #140B99 (Primary) 12D10 (Secondary) #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2107.05572

openalex publication_date 2021/07/07 · openalex created_date 2021/07/19 · openalex updated_date 2026/07/28

Abstract

We discuss the integer sequence transform a ↦ b where b_n is the number of real roots of the polynomial a₀ + a₁x + a₂x² + ⋯ + a_nxⁿ. It is shown that several sequences a give the trivial sequence b = (0,1,0,1, 0,1,…), i.e., b_n = n \bmod 2, among them the Catalan numbers, central binomial coefficients, n! and \binomn+kn 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 ≥ 3. Further, general procedures are given for constructing real sequences a_n for which b_n is either always maximal or minimal.

Citations

• The on-line encyclopedia of integer sequences

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