A common combinatorial description for a certain type of recurrences For integer-valued sequences $(x_n)_{n=0}^\infty$, consider recurrences of the form
$$x_n=ax_{n-1}+(bn+c)x_{n-2} \tag{$*$}\label{star}$$
for $n\ge2$, where $a,b,c$ are integers.
There seem to be many such sequences in the OEIS — see e.g. A000085, A001475, A005425, and A000898 — with various combinatorial descriptions.

Question: Is there a common combinatorial description for all integer-valued sequences $(x_n)_{n=0}^\infty$ satisfying recurrences of the form \eqref{star} — at least for natural $a$, $b$ ,$c$ and for some initial conditions (on $x_0$, $x_1$, depending on $a$, $b$, $c$)?

 A: So, the Fibonacci numbers can be constructed from this recursion, so it is natural to look for generalizations of those (and combinatorial models for the Fibonacci numbers).
The Fibonacci number $f_n$ (up to some index shift), can be seen as the number of integer compositions  of $n$ (list of numbers summing to $n$), using only the numbers $1$ and $2$.
Now, the recursion above, $x_n = a x_{n-1} + (bn+c)x_{n-2}$
hints that we also look at integer compositions using $1$ and $2$, but the $1$ comes in $a$ different colors.
Moreover, the 2s have two types, either, one of $c$ colors,
or it is a 'special' 2, with one of $b$ colors and a label.
The label is between 1 and the total sum of entries up to that point.
For example, $a=3$, $b=3$, $c=2$ $n=8$ has the object
$$
1_3, \; 2_{1, 1}, \; 2_{3, 5},  \; 2_{2}, \; 1_{1}.
$$
The numbers sum to $n$, and subscripts of the $1$ is between $1$ and $a$. The $2's$ with only one subscript has
the subscript between $1$ and $c$. Finally, each $2$
with two substripts, has the first subscript (color) between $1$ and $b$. The second subscript must be between $1$
and the sum of all numbers in the list up to that point.
For example, for $2_{3, 5}$, the 5 is the maximum allowed since $1+2+2=5$.
Below is Mathematica code for generating such compositions:
a = 3;
b = 3;
c = 2;
Clear[f];
f[0] := {{}};
f[1] := Table[{{1, j}}, {j, a}];
f[n_] := f[n] = Join[
    Join @@ Table[
      Append[f, {1, aa}],
      {f, f[n - 1]}, {aa, a}]
    ,
    Join @@ Table[
      Append[f, {2, bb}],
      {f, f[n - 2]}, {bb, b}]
    ,
    Join @@ (Join @@ Table[
        Append[f, {2, cc, j}],
        {f, f[n - 2]}, {cc, c}, {j, n}])
    ];

The natural initial condition is $x_0 =0$ and $x_1 = a$. (In the code above, I use $f$ instead of $x$)
