We are going to expand this:

$$({e}^{x})_{n}=\int_{0}^{\nu} \mu \,	\cdots \int_{0}^{\gamma} \beta \, \int_{0}^{\beta} \alpha \, \int_{0}^{\alpha} x \ {e}^{x} \,dx\,d\alpha\,d\beta\,d\gamma\cdots\,d\mu$$
where the integral symbol occurs $n$ times 

So we have (you can check by yourself)

$$
\begin{split}
({e}^{x})_{1} &  = {e}^{\alpha}\left(\alpha-1\right)+1 \\
({e}^{x})_{2} &  = {e}^{\beta}\left({\beta}^2-3{\beta}+3\right) + \left(\frac{{\beta}^2}{2}-3\right) \\
({e}^{x})_{3} &  = {e}^{\gamma}\left({\gamma}^3-6{\gamma}^2+15{\gamma}-15\right) + \left(\frac{{\gamma}^4}{2.4}-3\frac{{\gamma}^2}{2}+15\right)\\
 \end{split}
$$
and so on...<br>

The general pattern is as follows
$$
({e}^{x})_{n}={e}^{\nu}\sum_{k=0}^{n}(-1)^{k}a(n,k){\nu}^{n-k}+\sum_{k=0}^{n-1}(-1)^{k}a(k+1,k+1)\frac{{\nu}^{2(n-k-1)}}{(2(n-k-1))!!}
$$

The coefficient $a(n,k)$ is called a Bessel coefficient (https://oeis.org/A001498).

While we have also
$$({e}^{x})_{n}=\int_{0}^{\nu} \mu \,	\cdots \int_{0}^{\gamma} \beta \, \int_{0}^{\beta} \alpha \, \int_{0}^{\alpha} x \ \left(\sum_{k=0}^{\infty}\frac{{x}^{k}}{k!}\right) \,dx\,d\alpha\,d\beta\,d\gamma\cdots\,d\mu=\sum_{k=0}^{\infty}\frac{{\nu}^{k+2n}}{k!\prod_{i=1}^{n}(k+2i)}$$

So equating we get:
$$
P_{n}(\nu)=\sum_{k=0}^{n}(-1)^{k}a(n,k)\frac{1}{{\nu}^{n+k}}+{e}^{-\nu}\sum_{k=0}^{n-1}(-1)^{k}\frac{a(k+1,k+1)}{(2(n-k-1))!!}\frac{1}{{\nu}^{2(k+1)}}
$$