I'm trying to solve the following recurrence relation for $\alpha_j$, for which Mathematica is not helpful.
$$ \lambda\alpha_j + (j+1)\alpha_{j+1} = \sum_{\mu = 0}^{j}\frac{\mu(j-1)!}{\mu!(j-\mu)!}\alpha_{\mu} $$
Don't be deterred by the inclusion of the extra parameter $\lambda$. Any value of $\lambda$ is interesting for my purposes, so if somebody were to show me a solution even for a particular $\lambda$, I would be ecstatic :). I am by no means expecting a solution for arbitrary $\lambda$. $\alpha_0$ will clearly be left hanging, so don't worry about that, either.
I can do the first few $\alpha_j$ out by hand, so it seems solvable, but I'm wondering if there is an analytic method that I could use for future like problems also.
In short, anything helps! Thank you!