<s>[INST] Do the coefficients of these polynomials alternate in sign? Define polynomials $p_i(x)$ by the recurrence
\begin{align}
p_0(x)&=0 \\
p_1(x)&=1 \\
p_{2i}(x)&=p_{2i-1}(x)-p_{2i-2}(x) \\
p_{2i+1}(x)&=xp_{2i}(x)-p_{2i-1}(x) \\
\end{align}
The first few are given by
\begin{align}
p_0(x)&=0\\
p_1(x)&=1\\
p_2(x)&=1\\
p_3(x)&=x-1\\
p_4(x)&=x-2\\
p_5(x)&=x^2-3x+1\\
p_6(x)&=x^2-4x+3\\
p_7(x)&=x^3-5x^2+6x-1\\
p_8(x)&=x^3-6x^2+10x-4
\end{align}
It is reasonable to ask whether the coefficients of these polynomials alternate in sign. Any thoughts?
 [/INST] I reckon that
$$p_{2n-1}(x)=\sum_{k=0}^{n-1}(-1)^k \binom{n+1-k}k x^{n-1-k}$$
and
$$p_{2n}(x)=\sum_{k=0}^{n-1}(-1)^k \binom{n+2-k}k x^{n-1-k}$$
and that these can be proved by induction.
</s>