Consider the sequence of [Apéry numbers](http://oeis.org/A005259)  

$$
A\_n = \sum\_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum\_{j=0}^k \binom{k}{j}^3
= \sum\_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2 .
$$
In an email, physicist [Alan Sokal](http://www.physics.nyu.edu/faculty/sokal/) conjectures that it is a [Stieltjes moment sequence](http://en.wikipedia.org/wiki/Stieltjes_moment_problem). That is, that there exists a probability measure $\mu$ on $[0,+\infty)$ so that
$$
A\_n = \int\_{[0,+\infty)} s^n\;d\mu(s)
\tag{1}$$
for $n = 0,1,2,\dots$. [Of course you can equivalently say that $\mu$ is a nondecreasing function with $\mu(0)=0$ and $\lim\_{x\to+\infty} \mu(s) = 1$ and that (1) is a Stieltjes integral, rather than a "measure" integral.] 

**Is that conjecture correct? Is $A\_n$ a Stieltjes moment sequence?** 

[This question is a follow-up to [A conjectured formula for Apéry numbers](https://mathoverflow.net/questions/178790/) , where a formula for $A\_n$ was established.]