Probably this is a trivial question, but I am unable to find an answer: is there a function $v(x)$ such that 
$$
\int\_{0}^\infty x^n e^{v(x)} dx =\frac{1}{n!}
$$
for all positiv integer n?



Integral transform and $\frac{1}{n!}$.
Your question is a special case of the *Hamburger moment problem*: given a sequence of positive numbers $(\mu\_n)\_{n\geq 0}$ decides if there exists a positive measure $\mu$ on $\mathbb{R}$ such that $\newcommand{\bR}{\mathbb{R}}$


$$\mu\_n=\int\_{\bR} x^n \mu(|dx|),\;\;\forall n=0,1,2,\dotsc. $$


There exist many necessary and sufficient conditions for the existence of such a measure. The classical text *The Laplace Transform* by D.V. Widder is a good place to look, especially Section 10-13 of Chapter III. [Here is a link to the book.](http://archive.org/details/laplacetransform031816mbp)


**Addendum 1.** If you do not care about the positivity of the measure $\mu$ then we have the following result of R.P. Boas. (See Section 14, Chapter III of Widder's book)


For any sequence of real numbers $(\mu\_n)\_{n\geq 0}$ there exists a signed measure $\mu$ on $[0,\infty)$ with the following properties


$$ \mu\_n=\int\_0^\infty x^n \mu(dx) ,\;\;\forall n=0,1,2,\dotsc, \tag{1} $$


and 


$$\int\_0^\infty|\mu|(dx)<\infty.\tag{2} $$


Recall that any signed measure $\mu$ is the difference of two positive measures $\mu=\mu^+-\mu^-$ and the total variation mesure is $|\mu|=\mu^++\mu^-$. On the semiaxis $[0,\infty)$ a signed measure $\mu$ has the from $\mu=d\alpha$, where $\alpha$ is a function of bounded variation on compact intervals.


**Addendum 2.** Consider the Fourier transform of the measure $\mu$ in **Addendum 1**. We have $\newcommand{\ii}{\boldsymbol{i}}$


$$ \widehat{\mu}(\xi):=\int\_{\bR} e^{-\ii\xi x} \mu(dx) =\sum\_{n\geq 0}\mu\_n\frac{(-\ii\xi)^n}{n!}.\tag{3} $$


There is an obvious problem with the above equality: the series in the right-hand side may not be convergent for all $\xi$ if the momenta $\mu\_n$ grow too fast. In fact if the momenta grow fast, there exist at least two measures $\mu$, $\mu'$ satisfying both (1) and (2) above. In your case the $\mu\_n$ decay very fast and my guess is that $\mu$ is unique. (The series in the right-hand side of (3) converges for any $\xi$ so it defines a continuous function $f(\xi)$ which can be viewed as the Fourier transform of $\mu$ in the sense of distributions. Now use the Fourier inversion formula to recover $\mu$.)


In any case, the space of solutions of (1) (2) is completely understood. A good place to look is Akhiezer book *The Classical Moment problem* or Chapter 16 in the book *Unbounded selfadjoint operators on Hilbert space* by Konrad Schmudgen. The story is quite rich.


