How does the author arrive at the last equality in the first line, i.e.$$\text{why is } [x^k]\frac{1}{1-y(1+x)} = \frac{1}{1-y}[x^k]\frac{1}{1-\left(\frac{y}{1-y}\right)x} \text{?}$$
2026-03-31 00:02:50.1774915370
Question regarding an algebraic manipulation in GFology
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The author factorised the denominator.
$$\left(1-y\right)\left(1-\left(\frac{y}{1-y}\right)x\right) = 1-y-yx = 1-y(1+x)$$
The coefficient of $x^k$ is unaffected by factoring out a function of $y$.