Lattice paths inside a table: Rows and columns linear combinations

A lattice path inside the $m\times n$ table $T$ is a sequence ${\nu }_1,\dots ,{\nu }_k$ of cells such that ${\nu }_{j+1}-{\nu }_j\in \{(1,-1),(1,0),(1,1)\}$ for all $j=1,\dots ,k-1$. The number of lattice paths in $T$ from the first column to the $(x,y)$-cell is written into that cell. We present a precise description of the minimal linear recurrences among rows, columns, and columns sums. As a result, we obtain several formulas for the number of all lattice paths from the first column to the last column of $T$, that is, the $n^{th}$ column sum. Our methods are based on three classes of operators, which will also be studied independently.