What is known about explicit (not necessarily closed-form) solutions to the recurrence $$R^n_k= (\alpha n) R^{n-1}_k + (\alpha' n + \beta' k) R^{n-1} _{k-1},$$ with initial condition $R_0^0 = 1$ and with $R^n_k = 0$ for $n < 0$ or $k < 0$? Special cases of this are closely related to recurrences satisfied by some interesting combinatorial numbers, such as the binomial coefficients and the Stirling numbers.

The more general recurrence $$R^n_k= (\alpha n + \beta k + \gamma) R^{n-1}_k + (\alpha' n + \beta' k + \gamma') R^{n-1} _{k-1},$$

is open Problem 6.94 in *Concrete Mathematics* (2nd edition, p. 319).

The closest published result I have found thus far is the following formula due to Neuwirth ("Recursively defined combinatorial functions: Extending Galton's board," *Discrete Mathematics*, 2001) for the case $\alpha' = 0$ of the *Concrete Mathematics* problem,

$$R^n_k = \prod_{i=1}^k (\beta' i + \gamma') \sum_{i=0}^n \sum_{j=0}^n s^n_i \binom{i}{j} S^j_k \alpha^{n-i} (\gamma - \alpha)^{i-j} \beta^{j-k},$$

which, of course, gives me an answer to my question when $\alpha'=0$. (Here, $s^n_i$ and $S^j_k$ are unsigned Stirling numbers of the first and second kinds, respectively.)

I have tried generating functions without any success thus far. An answer like Neuwirth's that involves sums and binomial coefficients or Stirling numbers would be fine, as would a partial answer or just another idea to try.