Lattice paths inside a table, II

Consider an $m\times n$ table $T$ and latices paths ${\nu }_1,\dots ,{\nu }_k$ in $T$ such that each step ${\nu }_{i+1}-{\nu }_i=(1,1)$, $(1,0)$ or $(1,-1)$. The number of paths from the $(1,i)$-blank (resp. first column) to the $(s,t)$-blank is denoted by ${\mathcal{D}}^i(s,t)$ (resp. $\mathcal{D}(s,t)$). Also, the number of all paths form the first column to the las column is denoted by ${\mathcal{I}}_m(n)$. We give explicit formulas for the numbers ${\mathcal{D}}^1(s,t)$ and $\mathcal{D}(s,t)$.