Triangular Arrays using context-free grammar

In this work, the Hao grammar $G=\{u\to u^{b_1+b_2+1}{v}^{a_1+a_2},v\to u^{b_2}{v}^{a_2+1}\},$ together with the correspondence between grammars and combinatorial differential equations, is employed to obtain an interpretation of any triangular array of the form $$T(n,k)=(a_{2}n+a_{1}k+a_0)T(n-1,k)+(b_{2}n+b_{1}k+b_0)T(n-1,k-1)\mathrm{.}$$ This lead to have an interpretation of $T(n,k)$ as an increasing tree. Explicit formulas and structural properties are then derived through analytic differential equations. In particular, the $r$-Whitney-Eulerian numbers and the cases where $b_{2}n+b_{1}k+b_0=1$ are obtained explicitly. \noindent Applications include new interpretation formulas for the $r$-Eulerian numbers with generating functions. We also obtain full generating functions for the case $a_2=-a_1$ using this approach.