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There is a fairly simple pattern to it.

$$1 y + $$

$$(1 + 1x)y^2+ $$

$$(1+1x+1x^2 + 1x^3)y^3 + $$

$$(1+1x+\dots+1x^7)y^4 + $$

$$(1+1x+\dots+1x^{15})y^5 + $$

$$\dots$$

Does anyone know of a way to get the closed form for this? In other words, a closed form for

$$1 + x + x^2 + x^3 + \dots$$

is

$$\frac{1}{1-x}$$

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6  
Using $1+x+x^2 + ... x^k=\frac{x^{k+1}-1}{x-1}$, we can rewrite this as $\frac{y}{x-1}\sum_{n=0}^\infty (x^{2^n}-1)y^{n}$. This is only going to have a closed form if $\sum_n x^{2^n}y^{n}$ has a closed form. –  Thomas Andrews Jan 31 '12 at 14:07
    
@Matt Have you tried anything new on this? –  Pedro Tamaroff Mar 8 '12 at 4:06
    
Where does this come from? What exactly do you need from the generating function? –  vonbrand Feb 10 '13 at 2:17
    
@vonbrand: I'm not really looking for that solution anymore, but I was interested in manipulating other generating functions with it. –  Matt Groff Feb 19 '13 at 20:15
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