What is this polynomial series?

Let $F_0(x) = \text{Ei}(x)$.
It seems $F_n(x) = \text{Ei}(x) \frac{x^n}{n!} - \frac{e^x}{n!} P_n(x)$ where $P_n$ is a polynomial of degree $n-1$ for $n \ge 1$ and $F_n'(x) = F_{n-1}(x)$, which translates to $$ P_n'(x) + P_n(x) = P_{n-1}(x) + x^{n-1}$$ I get $$ \eqalign{P_1(x) &= 1\cr P_2(x) &= x+1\cr P_3(x) &= {x}^{2}+x+2\cr P_4(x) &= {x}^{3}+{x}^{2}+2\,x+6\cr P_5(x) &= {x}^{4}+{x}^{3}+2\,{x}^{2}+6\,x+24\cr P_6(x) &= {x}^{5}+{x}^{4}+2\,{x}^{3}+6\,{x}^{2}+24\,x+120\cr P_7(x) &= {x}^{6}+{x}^{5}+2\,{x}^{4}+6\,{x}^{3}+24\,{x}^{2}+120\,x+720\cr}$$

It seems $$P_n(x) = \sum_{m=0}^{n-1} (n-1-m)! x^m$$ which should be easy to prove by induction.

From the given polynomials, my guess is,

$$p_n(x)=\sum_{i=0}^n (n-i)!\cdot x^i$$

with the list starting from $n=0$