The Integer Sequence Transform $a \mapsto b$ where $b_n$ is the Number of Real Roots of the Polynomial $a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$

We discuss the integer sequence transform $a \mapsto b$ where $b_{n}$ is the number of real roots of the polynomial $a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{n} x^{n}$ . It is shown that several sequences $a$ give the trivial sequence $b = (0,1,0,1, 0,1,\ldots) $, i . e .,$ {b_n = n \bmod 2}$, among them the Catalan numbers, central binomial coefficients, $n!$ and $(\binom{n + k}{n})$ for a fixed $k$ . We also look at some sequences $a$ for which $b$ is more interesting such as $a_n = (n+1)^k $f o r$ k \geq 3$. Further, general procedures are given for constructing real sequences $a_{n}$ for which $b_{n}$ is either always maximal or minimal.

The Integer Sequence Transform $a \mapsto b$ where $b_{n}$ is the Number of Real Roots of the Polynomial $a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{n} x^{n}$

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The Integer Sequence Transform $a \mapsto b$ where $b_{n}$ is the Number of Real Roots of the Polynomial $a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{n} x^{n}$

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The Integer Sequence Transform a↦ba↦b where bnbn​ is the Number of Real Roots of the Polynomial a0+a1x+a2x2+⋯+anxna0​+a1​x+a2​x2+⋯+an​xn

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The Integer Sequence Transform a↦ba↦b where bnbn​ is the Number of Real Roots of the Polynomial a0+a1x+a2x2+⋯+anxna0​+a1​x+a2​x2+⋯+an​xn

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The Integer Sequence Transform $a \mapsto b$ where $b_{n}$ is the Number of Real Roots of the Polynomial $a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{n} x^{n}$

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The Integer Sequence Transform $a \mapsto b$ where $b_{n}$ is the Number of Real Roots of the Polynomial $a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{n} x^{n}$

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