Trying to answer this question, I found the following polynomials $$\left( \begin{array}{cc} n & P_n(x) \\ 0 & 1 \\ 1 & x \\ 2 & 2 x^2-x \\ 3 & 6 x^3-8 x^2+x \\ 4 & 24 x^4-58 x^3+22 x^2-x \\ 5 & 120 x^5-444 x^4+328 x^3-52 x^2+x \\ 6 & 720 x^6-3708 x^5+4400 x^4-1452 x^3+114 x^2-x \\ 7 & 5040 x^7-33984 x^6+58140 x^5-32120 x^4+5610 x^3-240 x^2+x \\ \end{array} \right)$$
The first and last coefficients are obvious; the other ones seem to be Eulerian numbers of different kinds.
What could they be ?
Edit
After @MartinR's comment, defining $$P_n(x)=(-1)^n\, Q_n(-x)$$ we face polynomials whose coefficients are the second-order Eulerian numbers $E_2(n, k)$ with $E_2(0, k) = \delta_{0, k}$ as said in $OEIS$ (sequence $A340556$).
Starting from this, I found this question relating the second-order EEulerian numbers and Lambert function (this is the origin of my problem). In his answer, @Marko Riedel proved that
$$\frac{1}{1+W(z)} = \sum_{m= 0}^\infty (-1)^m \,\frac{m^m}{m!}\,z^m$$
