What is a generating function? What is a generating function?
In the answer to this question this series comes up.
Its generating function is $$A(x) = \sum_{k\ge0} \frac{x^{4^k}}{1-x^{4^k}}$$
Which I took to mean the $x$th element of the series is given by $A(x)$
I guess that is not the case.  I tried to read the definition of a generating function but it's all Greek to me.  Is it simply that $A(x)$ sums to the next possible value in the series, but may not necessarily converge to a value?
 A: Strictly speaking, $\sum_{k\geq 0}\frac{x^{4^k}}{1-x^{4^k}}$ is not a power series, but it can be easily converted into something of the form $\sum_{n\geq 0} a_n x^n$. When $a_n$ is related with the number of objects with weight $n$ in a combinatorial context, we say that the power series $\sum_{n\geq 0}a_n x^n$ is a generating function.
The interesting part of analytic combinatorics comes when we prove that under certain assumptions we can derive the asymptotic behaviour of $a_n$ directly from the behaviour of its generating function.
Generating functions are formal objects, but if some bound of the form $a_n\leq C\cdot M^n$ holds, then $\sum_{n\geq 0}a_n x^n$ is uniformly convergent over any compact set contained in the region $|x|\leq\frac{1}{M}$.
A: The answer of the question depends on what kind of generating function is used to define the sequence $(a_n)$. It is not the same an ordinary generating function than an exponential generating function, by example.
An ordinary generating function have the usual form of a power series, that is
$$\sum_k a_k x^k$$
and a exponential generating function have a specific power series form
$$\sum_k a_n\frac{x^k}{k!}$$
A generating function can be an analytic function[*] such that it series expansion (ordinary or exponential) generates (hence it name) the sequence of coefficients $a_n$.
By example: the  exponential generating function of the Bernoulli numbers is defined by
$$\frac{x}{1-e^x}=\sum_{k=0}^\infty B_k \frac{x^k}{k!}$$
where in this case the coefficients $B_k$ are the Bernoulli numbers.
Other example: the ordinary generating function of the Fibonacci numbers is
$$\frac{x}{1-x-x^2}=\sum_{k=0}^\infty F_k x^k,\quad |x|<1$$
where the coefficients $F_k$ are the numbers (the sequence of) Fibonacci.
For what is useful a generating function? By example: some generating functions can be used to define a recursion for it coefficients $a_n$, you can see it in the free book generatingfunctionology of Wilf in page 22 where it is introduced the procedure "$x D \log$" to define these recursions.
[*]: I dont knew, just Im seeing now in the wikipedia article about generating functions that a generating function can be just formal, so it doesnt necessarily need to be convergent. In this case if the series diverges it (obviously) doesnt represent an analytical function.
A: In your example, it means
$$
\sum_{k=0}^\infty \frac{x^{4^k}}{1-x^{4^k}} = \sum_{n=0}^\infty a_n x^n
$$
where $(a_n)$ is the sequence of interest.  The function $A(x)$ is the "generating function" for the sequence $(a_n)$.
added
Expanding, we get
$$
A(x) = x+{x}^{2}+{x}^{3}+2\,{x}^{4}+{x}^{5}+{x}^{6}+{x}^{7}+2\,{x}^{8}+{x}^{
9}+{x}^{10}+{x}^{11}+2{x}^{12}+{x}^{13}+{x}^{14}+{x}^{15}+3{x}^{16
}+{x}^{17}+{x}^{18}+{x}^{19}+2{x}^{20}+{x}^{21}+{x}^{22}+{x}^{23}+2
{x}^{24}+\dots
$$
So, for example, $a_{20} = 2$ means that $2$ the coefficient of $x^{20}$.  Summing the series is not involved.
