Generating functions and sums with binomial coefficients

[burritoloco]

Homework Statement

Show that the generating function $A(x) = \sum_n a_n x^n$ of

$$a_n = \sum_{k=0}^n {n+k \choose 2k} 2^{n-k}$$

satisfies

$$A(x) = \frac{1-2x}{4x^2-5x+1}$$

Homework Equations

The Attempt at a Solution

A hint was given to "interchange the sums". After doing that, I don't see how to proceed. I also obtained the coefficients by partial fractions on A(x) but it's definitely non-trivial to show these are a_n. Thanks for any help.

 

[jbunniii]

Can you show what you got after interchanging the sums?

 

[burritoloco]

This one is done. Thanks for taking a look!