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The integer sequence transform <i>a</i> → <i>b</i>, where <i>b</i><sub>n</sub> is the number of real roots of the polynomial <i>a</i><sub>0</sub> + <i>a</i><sub>1</sub><i>x</i> + <i>a</i><sub>2</sub>x<sup>2</sup> + · · · + <i>a</i><sub>n</sub>x<sup>n</sup>
Abstract
We discuss the integer sequence transform a 1→ b, where bn is the number of real roots of the polynomial a0 + a1x + a2x2 + · · · + anxn. It is shown that several sequences a give the trivial sequence b = (0, 1, 0, 1, 0, 1, . . .), i.e., bn = 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 an = (n + 1)k for k ≥ 3. Further, general procedures are given for constructing real sequences an for which bn is either always maximal or minimal.
