Triangular arrays defined by the recurrence T(n,k) = (a2 n + a1 k + a0) T(n-1,k) + (b2 n + b1 k + b0) T(n-1,k-1) are interpreted as increasing trees via the Hao grammar, yielding explicit formulas for r-Whitney-Eulerian numbers and cases where b2 n + b1 k + b0 = 1.
Riehl,Category Theory in Context
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Triangular Arrays using context-free grammar
Triangular arrays defined by the recurrence T(n,k) = (a2 n + a1 k + a0) T(n-1,k) + (b2 n + b1 k + b0) T(n-1,k-1) are interpreted as increasing trees via the Hao grammar, yielding explicit formulas for r-Whitney-Eulerian numbers and cases where b2 n + b1 k + b0 = 1.