The integer sequence transform a → b, where b_n is the number of real roots of the polynomial a ₀ + a ₁ x + a ₂ x ² + · · · + a_nxⁿ

Abstract

We discuss the integer sequence transform a 1→ b, where bn is the number of real roots of the polynomial a ₀ + a ₁ x + a ₂ x ² + · · · + 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 mod 2, among them the Catalan numbers, central binomial coefficients, n! and 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.