<s>[INST] Polynomials $P(x)$ such that $P(x-1)$ $=$ $P(-x)$ Are there an infinite number of polynomials $P(x)$ of even degree such that 
$P(x-1) = P(-x)$
$P(x) = x^2+x+1$ is a good example because $P(x-1) = (x-1)^2+(x-1)+1 = x^2-x+1 = P(-x)$.
$P(x) = x^4+2x^3+4x^2+3x+1$ is another example because $P(x-1) = (x-1)^4+2(x-1)^3+4(x-1)^2+3(x-1)+1 = x^4-2x^3+4x^2-3x+1 = P(-x)$.
Is there an easy way to generate such polynomials? Thanks for help. 
 [/INST] The condition $P(-x)=P(x-1)$ is equivalent to $P(x-1/2)=P(-1/2-x)$,
namely that $P(x-1/2)$ is an even function. That should be helpful.
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