How to solve the following recurrence relations?

Question

I encounter the following recurrence relations when trying to find a general formula for the derivative of $f(x) = (1 + e^x)^{-1}$ ,

for $n\geq 0$,

\begin{align*} a(0,1) &= 1,\\ a(n,0) &= 0, \quad \\ a(n+1,k) &= (k - 1)a(n, k - 1) - a(n,k),\quad 1 \leq k \leq n+1, \\ a(n + 1,n+2) &= (n+1)a(n,n+1), \end{align*} For $n \geq 0$, according to that post, the solution seems to be $$a(n,k) = (-1)^n \sum_{j = 0}^{k - 1}(-1)^j\binom{k - 1}{j}(j+1)^n,\quad 0 \leq k \leq n+1$$ but I don't know how to get there. I never encounter recurrence relation of this form before.

So here are my questions.

1. Is there any method to solve this type of recurrence? If any, would you mind to tell me the book/reference on this topic might be.
2. I'm not sure about this, but is the solution to the problem unique? I think it is, by tring to apply the relations.

Most importantly, how to find the solution above? Any help would be appreciated.

Answer 1

If I had to solve this (I don't have to since you are asking for a method, not a solution), I would use a two-variable generating function In the form

$A(x, y) =\sum_{i=0}^{\infty}\sum_{j=0}^{\infty} a(i,j)x^iy^j $

and see what properties of A follow from the recurrence.