Generating function of the succession $\{\frac{n^{3}}{n!}\}_{n\in\mathbb{N}}$

Question

I'm having serious trouble with this question. What is the generating function of $\{\frac{n^{3}}{n!}\}_{n\in\mathbb{N}}$? I have thought first that I know some things related to this. Firstly, I know that the generating function of $\{\frac{1}{n!}\}_{n\in\mathbb{N}}$ is $e^{x}$, just using Taylor. Secondly, I also know that the generating function of $\{n^{3}\}_{n\in\mathbb{N}}$ is $\left(x\frac{\delta}{\delta x}\right)^{3}\left(\frac{1}{1-x}\right)=\frac{x^3+4x^2+x}{(1-x)^{4}}$. Nevertheless, multiplying this two generating functions doesn't give me the one that I am looking for... After this point, I have ran out of valid ideas... a little hint would be extremely appreciated! Thanks in advanced!