A straightforward formula: $f_{n,k}$ is the coefficient of $x^n$ in $$\frac{n!}{k!} \cdot \left( \frac{1^{-1}x^1}{1!} + \frac{2^0x^2}{2!} + \ldots + \frac{n^{n-2}x^n}{n!}\right)^k$$

A straightforward formula: $f_{n,k}$ is the coefficient of $x^n$ in $$\frac{n!}{k!}\cdot\left( \sum_{i=1}^{\infty} \frac{i^{i-2}x^i}{i!}\right)^k$$ (clearly, the sum here can be restricted to first $n$ terms).