Recurrence Relation derivatives of $x^x$

Question

I was recently trying to derive explicit formulas for the nth derivative of $f(x) = x^x$ which boiled down to solving the following recurrence relation (or something similar if I have made algebraic mistakes, which is fairly likely, this is besides the point though; I don't care that much about the derivatives themselves): $$\begin{Bmatrix} n+1\\ k\\ \end{Bmatrix} = (1-k)\begin{Bmatrix} n\\ k-1\\ \end{Bmatrix} + \frac d{da}\begin{Bmatrix} n\\ k-1\\ \end{Bmatrix} +a\begin{Bmatrix} n\\ k\\ \end{Bmatrix} $$ where the coefficient $\begin{Bmatrix} n\\ k\\ \end{Bmatrix}$ corresponds to the coefficient for the $x^{x-k}$ term in the nth derivative of $x^x$. Each of these coefficients is a function of $a, n $ and $k$. I would like to find an explicit formula for $\begin{Bmatrix} n\\ k\\ \end{Bmatrix}$. I also know that for $0>k>n$, $\begin{Bmatrix} n\\ k\\ \end{Bmatrix}=0$, and that $\begin{Bmatrix} n\\ 0\\ \end{Bmatrix}=a^n$ (for all reasonable n, that is, non-negative integers). I believe this is sufficient (or perhaps even redundant or excessive) information for the solution of this problem. Unfortunately, I am unsure how to go about doing this...Thanks!