Triangular Arrays using context-free grammar

In this work, the Hao grammar $G=\{\, u\rightarrow u^{b_1+b_2+1} v^{a_1+a_2},\quad v\rightarrow u^{b_2}v^{a_2+1} \,\},$ together with the correspondence between grammars and combinatorial differential equations, is employed to obtain an interpretation of any triangular array of the form $$T(n,k)=(a_2 n + a_1 k + a_0)\,T(n-1,k) + (b_2 n + b_1 k + b_0)\,T(n-1,k-1).$$ 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 $b_2n+b_1k+b_0=1$ 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.