TITLE: Is there a Definite Integral Representation for $n^n$?
QUESTION [9 upvotes]: The factorial $n!$ has a nice representation as definite integral:
$$
    n!=\Gamma(n+1)=\int_0^\infty t^{n} e^{-t}\, \mathrm{d}t. \!  
$$
Is it possible to write down such an integral for $n^n$ as well?
I tried to come up with an integral that reproduces a $n$ factor, $n$-times, but without success. I don't see a way to stop the partial integration process like in the $n!$ case. So this might not work here and I currently can't think of another way. If it helps to restrict $n$, feel free to do so.
The only thing a found online so far is the Lambert's $W$ function, which is involved when solving $x^x=z$, but I'm not sure if this helps. 
EDIT: Answers with integrals of the form $\displaystyle n^n=\int_0^\infty \cdots dt$ are preferred.

REPLY [2 votes]: The correct question (which someone has just asked me) may be as follows:
Find $f(t)$ (positive and independent of $n$) such that
$$n^n = \int_0^\infty t^n f(t) \,dt,\quad \forall\,n$$
That is, find $f(t)$ such that $\{n^n\}$ is the sequence of moments of f(t)dt.
This is a particular instance of the Stieltjes moment problem.
There is criterion (due to Carleman) which guarantees that such
a solution is unique, but I don't know an explicit form for $f(t)$.