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