TITLE: What is this polynomial series?
QUESTION [6 upvotes]: I've come across this simple, yet peculiar series of polynomials wich I can't quite find a general formula for.
It goes as such:
$$1$$
$$x+1$$
$$x^2+x+2$$
$$x^3+x^2+2x+6$$
$$x^4+x^3+2x^2+6x+24$$
$$\vdots$$
I'm finding these polynomials by looking at a term in the $n^{th}$ integral of Ei(x).
For example, the 5th integral of Ei(x) is;
$$\frac{1}{120}(x^5Ei(x)-e^x(x^4+x^3+2x^2+6x+24))$$

REPLY [6 votes]: 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.

REPLY [3 votes]: From the given polynomials, my guess is,
$$p_n(x)=\sum_{i=0}^n (n-i)!\cdot x^i$$
with the list starting from $n=0$