How can I find a closed form of the ordinary generating function for sequences

Question

I have to find a closed form for the sequences: $a_{n}=-a_{n-1}+5a_{n-2}-3a_{n-3}$ with $a_{0}=7$, $a_{1}=-12$ and $a_{2}=49$ I know that I have to multimply it with $z^n$. I have than $\sum a_{n}z^n=-\sum a_{n-1}z^n+5\sum a_{n-2}z^n-3\sum a_{n-3}z^n$ and $\sum a_{n}z^n=-z\sum a_{n-1}z^{n-1}+5z^2\sum a_{n-2}z^{n-2}-3z^3\sum a_{n-3}z^{n-3}$ How can I do it then

Answer 1

You have the right idea but should be explicit with the summation indices: $$\sum_{n=3}^\infty a_n z^n = -z\sum_{n=3}^\infty a_{n-1}z^{n-1}+ 5 z^2\sum_{n=3}^\infty a_{n-2}z^{n-2} - 3z^3\sum_{n=3}^\infty a_{n-3}z^{n-3}$$ The next step is to express these sums in terms of the generating function $A(z)=\sum_{n=0}^\infty a_n z^n$, as follows: $$A(z) - a_0 - a_1z -a_2 z^2 = -z\left(A(z)-a_0 - a_1 z \right)+ 5 z^2\left(A(z) - a_0\right) - 3z^3 A(z)$$ Now substitute the known values of $a_0$, $a_1$, and $a_2$, and solve for $A(z)$.