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 $(T(n,k))$ 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 $a_{n,k}=a_0+a_{1}k+a_{2}n$ and $b_{n,k}=b_0+b_{1}k+b_{2}n$. 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 Faà 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. $\mathbf{\text{Keywords:}}$ triangular recurrence, weighted paths, $r$-Eulerian numbers, combinatorial interpretation.