Approach to simplify using Partial Fraction Decomposition

Question

Can anyone suggest an approach to simplify this closed form generating function using partial fraction decomposition?

Thanks in advance!

Answer 1

As Mark Bennet said, you need to factorize the denominator. Once this will be done, you should easily arrive to $$-\frac{14}{2 z+1}-\frac{20}{3 (4 z-1)}+\frac{4}{(2 z+1)^2}-\frac{4}{(4 z-1)^2}-\frac{13}{3 (z-1)}+\frac{3}{(z-1)^2}$$