# Homework Help: Generating functions and sums with binomial coefficients

1. Mar 1, 2012

### burritoloco

1. The problem statement, all variables and given/known data
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

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

2. Relevant equations

3. 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.

Last edited: Mar 1, 2012
2. Mar 1, 2012

### jbunniii

Can you show what you got after interchanging the sums?

3. Mar 1, 2012

### burritoloco

This one is done. Thanks for taking a look!