<s>[INST] Finding generating series from equation How can I solve $f(x)=(1+x+x^2)f(x^2)$, where $f$ is a generating series of a set?  I just need the technique, not necessarily the solution.
 [/INST] $f(x) = a_0 + a_1 x + a_2 x^2 + \cdots$
$f(x^2) = a_0 + a_1 x^2 + a_2 x^4 + \cdots$
so $(1 + x + x^2) f(x^2) = a_0 + a_1 x + a_2 x^2 + \cdots + a_0 x + a_1 x^2 + a_2 x^3 + \cdots + a_0 x^2 + a_1 x^3 + a_2 x^4 + \cdots$
Given $f(x) = (1 + x + x^2) f(x^2)$ we equate the following powers of $x$:
$x^0$ therefore:  $a_0 = a_0$
$x^1$ therefore:  $a_1 = a_1 + a_0$
But these two equations shows that $a_0 = 0$ ($\checkmark$).
$x^2$ therefore:  $a_2 = a_2 + a_1 + \underbrace{a_0}_{=0}$
Thus $a_1 = 0$ ($\checkmark$).
Can you continue?
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