Answering a recent question I came across the family of polynomials:
$$P_n(x)=\sum_{k=0}^n\binom{n+k}{2k}(-x)^k$$
with numerical evidence of the following interesting properties:
- $P_n(2)=\begin{cases}+1,& n=0,3\mod4\\-1,&n=1,2\mod4\end{cases}$
- $P_n(4)=(-1)^n(2n+1)$
- all roots are simple and real and belong to the interval $(0,4)$
- the absolute value of the polynomial on $(0,2)$ seems to be bounded by a value close to $\sqrt2$
The first two properties can be easily proved using the recurrence relation $$ P_{n+1}(x)=P_n(x)-x\sum_{j=0}^n P_j(x), $$ but I have no idea how to approach the other two. Any hint is appreciated. Additionally I would like to estimate the absolute value of the largest extremum (it is situated between the last two zeros).