I am in the middle of a development and I stumbled on recursively defined numbers. They don't ring a bell for me, but maybe they were already studied. Anybody has a hint?
$$ \kappa^{(k+1)}_{\delta} = \sum_{\beta = 1}^{\delta-k}\kappa^{(k)}_{\delta - \beta}\binom{\delta}{\beta} = \sum_{\beta' = k}^{\delta-1}\kappa^{(k)}_{\beta'}\binom{\delta}{\beta'} $$
with $\kappa^{(1)}_{\delta}=1$ for all $\delta$. At the beginning, they are pretty manageable. E.g. $$ \kappa^{(2)}_{\delta} = 2^{\delta} - 2 $$ $$ \kappa^{(3)}_{\delta} = 3(3^{\delta-1} - 2^{\delta} + 1 ) $$
I'm still working on finding a general solution for those numbers but I was wondering if they looked familiar to anybody.
A straightforward consequence of their definition is $$ \kappa^{(\delta)}_{\delta} = \delta\kappa^{(\delta - 1)}_{\delta - 1} = \delta! $$
I suspect they have the property $$ \sum_{k = 1}^{\delta} (-)^k\kappa^{(k)}_{\delta} = (-1)^\delta $$
edit: Those counts appear in the powers of mapping of subsets of a range $[0, D]$. You define $K^{(\alpha\beta)}$ with $\alpha > \beta$ as the function taking a pair of sets of size $\alpha$ and $\beta$ respectively and returning 1 when they intersect and 0 otherwise. Then you arrange all subsets of $[0, D]$ in a basis set so that you define $K$ as the matrix mapping all the subsets together (it is a triangular matrix). It is composed of many $K^{(\alpha\beta)}$ blocks. Those coefficient $\kappa^{(k)}_{\alpha-\beta}$ appear when you take the $k^{th}$ power of $K$. More precisely, using a kinda of notation shortcut $$ K = \sum_{\alpha = 2}^D\sum_{\beta=1}^{\alpha-1} K^{(\alpha\beta)} $$ and $$ K^k = \sum_{\alpha = k+1}^D\sum_{\beta=1}^{\alpha-k} \kappa^{(k)}_{\alpha-\beta} K^{(\alpha\beta)} $$
For two subsets $S, S'$ of the range $[0, D]$ with $S \subset S'$, $K^k(S, S')$ counts the number of ways you can split $S'\setminus S$ in $k$ subsets exactly.
