  We study the triangular array defined by the Graham--Knuth--Patashnik
recurrence $T(n,k) = (\alpha n + \beta k + \gamma)\, T(n-1,k)+(\alpha' n +
\beta' k + \gamma') \, T(n-1,k-1)$ with initial condition $T(0,k) =
\delta_{k0}$ and parameters $\mathbf{\mu} = (\alpha,\beta,\gamma,
\alpha',\beta',\gamma')$. We show that the family of arrays $T(\mathbf{\mu})$
is invariant under a 48-element discrete group isomorphic to $S_3 \times D_4$.
Our main result is to determine all parameter sets $\mathbf{\mu} \in
\mathbb{C}^6$ for which the ordinary generating function $f(x,t) =
\sum_{n,k=0}^\infty T(n,k) \, x^k t^n$ is given by a Stieltjes-type continued
fraction in $t$ with coefficients that are polynomials in $x$. We also exhibit
some special cases in which $f(x,t)$ is given by a Thron-type or Jacobi-type
continued fraction in $t$ with coefficients that are polynomials in $x$.
