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Polynomials Whose Coefficients Are Stern Numbers
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Karl Dilcher

Department of Mathematics and Statistics

Dalhousie University

Halifax, Nova Scotia B3H 4R2

Canada

Larry Ericksen

1212 Forest Drive

Millville, NJ 08332-2512

USA

**Abstract:**

The main object in this paper is the sequence of polynomials
*P*_{n}(*z*) that have Stern numbers as their
coefficients; that is, the terms of Stern's diatomic sequence. We
derive certain basic properties of these polynomials, investigate
the distribution of their real and complex zeros, and prove some
results concerning factorizations and resultants. We also consider the
(0,1)-polynomials obtained from *P*_{n}(*z*)
by taking their coefficients modulo 2. In spite of its simple form, the
polynomial sequence is shown to possess some interesting algebraic and
analytic properties. Finally, we discuss combinatorial interpretations
of the polynomials *P*_{n}(*z*) and indicate
ways of generalizing them.

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(Concerned with sequences
A000045
A001045
A002487
A011655
A048573
A174868.)

Received July 14 2021; revised version received November 24 2021.
Published in *Journal of Integer Sequences*,
November 25 2021.

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