My question is a bit imprecise - but I hope you like it. I even strongly think it has a proper answer.

The binomial coefficient $\binom{\frac{1}{2}}{n}$ is strongly related to Catalan numbers - the expression $(1-4x)^{\frac{1}{2}}$ appears when calculating the generating function of the Catalan numbers and solving a quadratic equation.

I am trying to find some combinatorial interpretation of $\binom{\frac{1}{k}}{n}$ for non-zero integer $k$. I feel it must exist - I don't know if it is because of intuition or because I've seen something similar and forgot. I want an elementary interpretation, maybe related to trees (since Catalan numbers count binary trees).

So, can anyone find a combinatorial interpretation of those coefficients (possibly multiplied by some power such as $k^n$)?

My motivation: I can show p-adically that $\binom{\frac{1}{k}}{n}$ is $p$-integral for any prime $p$ not dividing $k$. I am looking for a combinatorial proof of this property, and a combinatorial interpretation of $k^m \binom{\frac{1}{k}}{n}$ (for some integer $m$) will suffice for this.