While working a combinatorial problem, I have encountered the recurrence relation $$a(n+1, k) = ka(n,k) + a(n,k-1)$$ where $a(0,0) = 1$ and $a(0,k)=0$ if $k \ne 0$.
Except for the $k$ multiplier, this would be the same recursion as the binomials.
Does anyone know about these numbers? Particularly if there is a non-recurring expression for them?
Some facts I've noted: $$a(0,k) = 0, k>0$$$$a(n,k)=0, n < 0\text{ or }k<0\text{ or } n < k$$ Restricting to $n \ge 1, 1 \le k \le n$
$$a(n,1)= a(n,n) = 1$$ $$a(n,2) = 2^{n-1} - 1$$ $$a(n,n-1) = \frac{n(n-1)}{2}$$ and it appears that $p \mid a(p, k), 2 \le k < p$ for prime $p$, though I've not confirmed it.