I have two series that I wanted to expand and to have the possibility to find the nth element for it. Using Wolfram software I was able to get the patterns but in generating function form. I tried to find some information and examples on how to find a closed-form from generating function but I do not quite understand how to do this except for several simple examples.
Here's the result I get for my series:
$G_n(a_n)(z) = (-626 z^3 + 90 z^2 + 45 z + 731)/((z - 1)^2 (z^2 + z + 1))$
and
$G_n(a_n)(z) = -(3 (27 z^2 - 20 z - 87))/((z - 1)^2 (z + 1))$
Any help is appreciated, thanks in advance.
One nice method: partial fractions. I will do one example.
$$ G(z) = \frac{-3( 27 z^2 - 20 z - 87)}{(z - 1)^2 (z + 1)} = \frac{30}{z+1} - \frac{111}{z-1} + \frac{120}{(z-1)^2} $$ Two are geometric series $$ \frac{30}{z+1} = \frac{30}{1-(-z)} = \sum_{n=0}^\infty 30 (-1)^n z^n \\ \frac{-111}{z-1} = \frac{111}{1-z} = \sum_{n=0}^\infty 111 z^n $$ and one is a binomial series $$ \frac{120}{(z-1)^2} = 120\;(1-z)^{-2} = 120\sum_{n=0}^\infty \binom{-2}{n}(-z)^n =\sum_{n=0}^\infty 120 (n+1) z^n $$ So the final result is $$ \sum_{n=0}^\infty \big(120n+231+30(-1)^n\big) z^n =261+321 z+501 z^2+561 z^3+741 z^4+801 z^5+\dots $$