What is the general method for finding the aymptotics of large $n$ of the sequence $(a_n)_{n=0}^\infty$ defined by the recursion
$$a_{n} = (\alpha_1n+\alpha_2) a_{n-1} + (\alpha_3n+\alpha_4) a_{n-2}+\delta \tag1$$
where $\alpha_i$'s are constant real numbers and $\delta\in\{0,1\}$ is constant.

Here is [an example][1] of the above recursion and my frustrated attempt at using the generating function for the un-simplified version. That particular example is solvable with a generating function once [it is transformed][2]. However, not every recursion of the form $(1)$ can be simplified through a simple transformation.


  [1]: https://math.stackexchange.com/q/3386637/64809
  [2]: https://math.stackexchange.com/a/3389236/64809