Any techniques for deriving a closed form solution for the following recurrence relation? Or bounds on asymptotic behavior for large $n$?

$$a_{n+1,k} = \sum_{0 \le i \le n} \frac{n!}{i!} a_{i,k-1}$$

where $n,k \ge 0$ and $a_{0,0} = 0$

  • 1
    $\begingroup$ You will need more boundary conditions; I'm guessing you'll want to set $a_{n, 0}$. $\endgroup$ – user44191 Feb 4 at 1:44
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    $\begingroup$ For fixed $r > 1$, if you set $a_{r-1,0} = 1$ and $a_{n,0} =0$ for $n\neq r$ then I believe the solution is given by the $r$-Stirling numbers $a_{n,k}=\left[\begin{array}{c} n \\ k+1 \end{array}\right]_r$. See Broder, "The r-Stirling numbers" (1984). $\endgroup$ – Timothy Budd Feb 4 at 8:31

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