Power series representation of the sum of $\sum_{k=0}^\infty(k^3+1) x^k/k!$ So, recently I was searching for, what seemed like a fairly simple one, what function does this sum represent:
$\displaystyle f(x)=\sum_{k=0}^\infty\left(k^3+1\right)\frac{x^k}{k!}$
So the natural thing would be to separate both equations:
$\displaystyle f(x)=\sum_{k=0}^\infty k^3\frac{x^k}{k!}+e^x$
So, with further identification $f(x)$ seemed like a simple answer:
$f(x)=T_3(x)e^x+e^x$
where $T_n$ is the $\,n^{\mathrm{th}}$ Touchard polynomial.
$f(x)=\left(x+3x^2+x^3+1\right)e^x$
What bugged me is the fact that neither I do I understand Touchard Polynomials nor does the question fit the level of knowing them, this is intended as a Second Semester undergrad Calculus II question, so my question is simple, is there a method of finding the sum without having to deal with the Touchard polynomials, is there a way of arriving to the same (or different, maybe I’m wrong) answer without dealing with them?

Note that
$$\begin{aligned}\frac{k^3+1}{k!}&=
\frac{k^2}{(k-1)!}+\frac1{k!}\\&
=\frac{(k+1)(k-1)+1}{(k-1)!}+\frac1{k!}\\&
=\frac{k+1}{(k-2)!}+\frac1{(k-1)!}+\frac1{k!}\\&
=\frac1{(k-3)!}+\frac3{(k-2)!}+\frac1{(k-1)!}+\frac1{k!}\end{aligned}$$
and that $\sum \frac{x^k}{(k-r)!}=\sum\frac{x^{k+r}}{k!}=x^re^x$. (Well, one has to be careful with the first summands of the series, but everything works out nicely)
