I have a table of values that follows a specific pattern. The right and bottom edge are all ones. Let the table be indexed from bottom up and rigth to left, i.e 0,0 is at bottom right. A value at position $i,j$ is the sum of the values to the bottom of it, to the right of it and diagonally behind it. See below: $$ \begin{matrix} 63 & 25 & 7 & 1 \\ 25 & 13 & 5 & 1 \\ 7 & 5 & 3 & 1 \\ 1 & 1 & 1 & 1 \end{matrix} $$ Expressed as a recurrence relation: $$S(i,j)=S(i-1,j-1) + S(i, j-1) + S(i-1, j)$$ $$S(0,j)=1 \ \forall j$$ $$S(i,0)=1 \ \forall i$$
How can I express $S(i,j)$ in a function? If possible.
Welcome to MSE!
If you want, you can solve this using generating functions. In particular, if we write
$$S(x,y) = \sum_{x,y} s(i,j) x^i y^j$$
we can use your recurrence to find
$$S(x,y) = \frac{1}{1 - (x + xy + y)}.$$
Then we can get a closed form for $s(i,j)$ by getting a closed form for the coefficients in a taylor expansion of $S$.
Thankfully, though, this has been done for us. We can just ask oeis for your sequence, and we find that it is the square array of "Delannoy Numbers", and the linked wikipedia article contains a variety of formulas for your expression. Unfortunately, there's nothing I would really call a "closed form", but the fact that there isn't one on wikipedia (or on oeis, for that matter) is good evidence that one simply doesn't exist. This will save us the time of actually looking for one.
I hope this helps ^_^