My goal is to learn how to solve recurrence functions consisting of two variables.
For example, I want to find how to solve the following function which I have found in a paper:
$$ T(d,a) = \begin{cases} 0 &\quad\text{if } d = a\\ \sum_{i=1}^{a} T_{drop}(d,i) + T\left(d-i,a-i+1\right) &\quad \text{if } d > a \\ \end{cases} $$
where
$T_{drop}(d,i) = \frac{\left( d - i +1 \right) \left( d - i + 2\right)}{2} $
The paper mentions that the generating function is
$$ G(x,y) = \left(1 -y(1=x)\right)^{-1} \cdot \sum_{0<i<j} T_{drop}(j,i)x^iy^j $$
The resulting closed form of $T(d,a)$ is
$$ T(d,a) = \sum_{i=0}^{a-1} \sum_{j=i}^{d-a+i-1} {j \choose i} \cdot T_{drop} (d-j,1) $$
I have just started learning on how to use generating functions to solve recurrences with 1 variable but I am having some trouble with this one. I am not sure how to approach this problem. Any help would be appreciated.