Explicit formulas and combinatorial interpretation of triangular arrays via weighted paths

**Abstract:** 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:

$$T(n, k) = a_{n,k}T(n - 1, k) + b_{n,k}T(n - 1, k - 1)$$

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 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.

**Keywords:** Combinatorial interpretation; Triangular recurrence; $r$-Eulerian numbers; Weighted paths

**2020 Mathematics Subject Classification:** 05A05; 05A15; 05A19