<s>[INST] Generating function for the sequence $1,1,3,3,5,5,7,7,9,9,\ldots$ The generating function for the sequence $\left\{1,1,1,1,...\right\}$ is $$1 + x + x^2 + x^3 ... = \frac{1}{1-x}$$
What is the generating function for the sequence $\left\{1,1,3,3,5,5,7,7,9,9,\dots \right\}$?

This is my attempt:
what we want to do is after the initial  $\left\{1,1,1,1,...\right\}$ function we want to add two more functions shifted two to the right such as $\left\{0,0,1,1,1,...\right\}$ and keep adding such function 2 at a time and each new two are shifted 2 more to the right then the previous one. so first would be $\left\{1,1,1,1,\dots \right\}$ then add 2 of $\left\{0,0,1,1,1,1,\dots\right\}$ then add 2 more of $\left\{0,0,0,0,1,1,1,1,\dots\right\}$ and keep doing that. this gives us the function 
$$\frac{1}{1-x}+\frac{2x^2}{1-x}+\frac{2x^4}{1-x}+...$$ or $$\frac{1}{1-x}+\frac{2x^{2k}}{1-x}$$ however $i$ do not know where to go from here, how do $i$ finish this problem?
 [/INST] You already did the difficult part... You probably mean $$\frac1{1-x}+\sum_{k=1}^\infty\frac{2x^{2k}}{1-x}$$
So you just have to evaluate the geometric series $$\sum_{k=1}^\infty (x^2)^k = \frac1{1-x^2} - 1 = \frac{x^2}{1-x^2}$$
And obtain
$$\frac1{1-x} + \frac2{1-x}\sum_{k=1}^\infty x^{2k} = \frac{1+\frac{2x^2}{1-x^2}}{1-x} = \frac{x^2+1}{(1-x)(1-x^2)}$$
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