For positive integers $n$, this family of polynomials seems to have interesting properties (i.e. irreducible over $\mathbb{ Z,Q}$, only real and negative roots,...)$$\frac{1}{(n+1) !} \sum_{k=0}^{\lfloor n / 2\rfloor} \frac{(2 n-k) !}{k !(n-2 k) !} X^{k}$$ Any idea where these come from?
Per Dietrich Burde's comment below I add the first few values \begin{align*} &1 \\ &X + 2\\ &5(X + 1) \\ &3X^2 + 21X + 14\\ &14(2X^2 + 6X + 3)\\ &6(2X^3 + 30X^2 + 55X + 22)\\ &33(5X^3 + 30X^2 + 39X + 13)\\ &55(X^4 + 26X^3 + 91X^2 + 91X + 26)\\ &143(7X^4 + 70X^3 + 168X^2 + 136X + 34)\\ &13(21X^5 + 840X^4 + 4760X^3 + 8568X^2 + 5814X + 1292)\\ &442(14X^5 + 210X^4 + 798X^3 + 1140X^2 + 665X + 133)\\ \end{align*}