I'm looking for the value of the generating function $$f_{k, n}(x) = \sum_{y \in [n]^k}{x^{\# \text{distinct coordinates of y}}} = \sum_i{a^{(k, n)}_ix^i}$$
For example, for $k = 1$, we have $$f_{1, n}(x) = nx,$$ and for $k = 2$, we have $f_{2, n}(x) = n(n-1)x^2 + nx$. I know that $$a_i^{(k+1, n)} = ia_i^{(k, n)} + (n - i + 1)a_{i - 1}^{(k, n)}.$$
Are there any closed formulas for the coefficients $a_i^{(k, n)}$, or any tight bounds on them in terms of n? I'm interested primarily in the case where n grows large and k is constant.
Thanks!
It appears that $a_i^{(k,n)}=S(k,i)(n)_i$, where $S(k,i)$ are the Stirling numbers of the second kind and $(n)_i=n(n-1)\cdots(n-i+1)$ is the falling factorial. I'm sure this would be a relatively easy induction exercise.