Fibonacci Generating Functions

The Fibonacci Sequence can be generated from the following recurrence relation: $$a(n)=a(n-1)+a(n-2) \text{ with } a(0)=1 \text{ and } a(1)=1$$ The generating function for this series is given by: $$\frac{x^2}{1-x-x^2}$$ How is generating function affected if we have a recurrence relation as follows: $$a(n)=a(n-1)+a(n-8)$$ I created a post about this titled Fibonacci-like Sequences quite recently on May 13th 2024. The new generating function is now: $$\frac{x^2}{1-x-x^8}$$ What if there are coefficients (let's say $\alpha$ and $\beta$ in front of the two terms on the LHS: $$a(n) = \alpha . a(n-1) + \beta . a(n-2)$$ In this case, the generating function becomes: $$\frac{x^2}{1-\alpha . x - \beta . x^2}$$ Let's take the following example: $$a(n)=4. \, a(n-1)+10. \, a(n-2)$$ The generating function becomes $$\frac{x^2}{1-4x - 10 x^2}$$ The sequence of terms becomes (permalink):

0, 1, 4, 26, 144, 836, 4784, 27496, 157824, 906256, 5203264, 29875616, 171535104, 984896576, 5654937344, 32468715136, 186424233984, 1070384087296, 6145778689024, 35286955629056

This numbers form the initial terms of OEIS A180226:

A180226

a(n) = 4*a(n-1) + 10*a(n-2), with a(1)=0 and a(2)=1.

To check what happens to the generating function when the tribonacci sequence is considered, see my post titled Beyond Fibonacci (March 10th 2019).