Generating functions formula

Question

Hi there I'm trying to establish a formula for $a_n$ using a generating function $$\sum_{n=0}^{\infty} a_n t^n = A(t)$$which I have worked out as $A(t)=\frac{1}{2(2t+1)}-\frac{1}{2(4t-1)}-\frac{1}{t+1}$

Does anyone know how to do this? Thanks

Answer 1

Since $$A(t)=\frac12\frac{1}{1+2t}+\frac12\frac{1}{1-4t}-\frac{1}{1+t},$$we obtain the desired series provided $|t|<\frac14$, whence$$a_n=\frac12(-2)^n+\frac124^n-(-1)^n.$$