While answering another question, I stumbled upon a system of equations, given the form:
$$C_{a,b,c}=\sum_{n=0}^{\max\{b,c\}}\left[\binom cn+(-1)^a\binom bn\right](-1)^nS_{a+n,b+c-n}$$
with known $C_{a,b,c}$. The motivation for finding when this is solvable is that it allows marking off some very nasty integrals as solvable.
I have, from where this came from, the values of $S_{0,q}$ and $S_{p,1}$. When $(a,b,c)=(2p,1,1)$, I end up with $C_{2p,1,1}=2S_{2p,2}-2S_{2p+1,1}$, and thus this is also solvable for $S_{2p,2}$.
It would also seem that this is solvable for $S_{p,8-p}$ except when $p=1,2,3$. From the above we know that $S_{6,2}$ and $S_{7,1}$ are known. We also have the equations:
$$\begin{bmatrix}C_{3,3,2}-S_{6,2}\\C_{2,4,2}-C_{2,3,3}-S_{6,2}\end{bmatrix}=\begin{bmatrix}1&-2\\1&-6\end{bmatrix}\begin{bmatrix}S_{4,4}\\S_{5,3}\end{bmatrix}$$
which lets us solve for $S_{4,4}$ and $S_{5,3}$. $S_{0,8}$ is also known.
I would love to spend the whole day trying to figure out which values are solvable, but frankly I don't think I'd make much progress. While expanding out the equations, I've noticed that whenever two equations start with the same two variables, they also have the same coefficients. However, I would imagine that as the arguments became larger, you would be able to solve more and more of the values for $S_{p,q}$ with small $q$.
Any ideas on which values we can solve for? Or perhaps moreso when this isn't solvable?
