Explicit Formulas and Combinatorial Interpretation of Triangular Arrays

Using the lattice paths in $\mathbb{N}\times\mathbb{N}$, we derive a general formula for sequences $\big(T(n,k)\big)$ satisfying the recurrence relation of the form: \begin{equation*} T((n,k)=a_{n,k}T(n-1,k)+b_{n,k}T(n-1,k-1). \end{equation*} We apply this result to the case where $a_{n,k}=a_0+a_1k+a_2n$ and $b_{n,k}=b_0+b_1k+b_2n$. This leads to explicit expressions for $T(n,k)$, with simpler formulas arising in the case $b_2=0$, as well as in the fully general case, using Fa\`a di Bruno's type expression. In particular, we analyze the case $b_{n,k}=1$, which frequently occurs in enumerative combinatorics. Applications include explicit formulas for the $r$-Eulerian numbers.We also express the case $b_{n,k}=1$, using a transition matrix. We apply our results to several sequences. \textbf{Keywords:} triangular recurrence, weighted paths, $r$-Eulerian numbers, combinatorial interpretation.