I am studying lower triangular matrix with elements:
$$ M_{i,j}=(-1)^j \frac{i!}{(i-j)!} F_{i-j} ,~ i\geq j,~ i,j=0,1,2,\dots$$
where $F_n$ are constants. It can be easily recursively inverted. However I wish to find an explicit formula depending on row/column number. It turns out that all columns of $M^{-1}$ can be expressed as function of the first column (index zero). However, this column has complicated structure where some patterns can be observed, but others I am unable to find. Here are two equivalent forms of the first column $M^{-1}_{i,0}$:
This is presumably a very hard question... but maybe I am lucky and somebody has an idea.