<s>[INST] Extracting sequence from generating function I was provided the following generating function, and was unsure how to use it. I have never seen an example where the function “involved” itself.
The generating function is
$F(z)^8$
Where
$$F(z)=z+z^6 F(z)^5+z^{11} F(z)^{10}+z^{16} F(z)^{15}+z^{21} F(z)^{20}$$
Any help appreciated
 [/INST] A method that will take a while but gives the first few terms for the equation
$$ F(x) = x + x^6 F(x)^5 + x^{11} F(x)^{10} + x^{16} F(x)^{15} + x^{21} F(x)^{20} $$
is to let $F(x) = a_{0} + a_{1} \, x + a_{2} \, x^2 + \cdots$.
First notice that the right-hand side does not have a constant term. This leads to $F(x)$ being $F(x) = x + a_{2} x^2 + \cdots$. Also notice that $a_2 = a_{3} = \cdots = a_{10} = 0$ and further reduces $F(x)$ to $F(x) = x + x^{11} + \cdots$
This leads to
\begin{align}
x +  x^{11} + a_{21} \, x^{21} + \cdots &= x + x^{11} (1 + 5 \, x^{10} + \cdots) + x^{21} (1 + 10 x^{10} + \cdots) + \cdots \\
&= x + x^{11} + (5 + 1) \, x^{21} + \cdots.
\end{align}
Continuing gives the result mentioned by Gary in the comments.
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