Understaing Concept of Generating function. I can easily understand the generating function of sequence such as $a_n = n+1$ or something, but cannot understand generating function for some object.
There are 2 examples (with photos):

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*What does "egf for counting set" mean? For example, $\{a_n\}=1,1,1,1,1…$   means numbers of set with $n$ elements or something? I can never understand.




*What is gf for prime path (in Schroder number) The number of prime paths? According to which one? I am so confused.


I think they are very basic questions. I'm sorry for that
 A: $1-)$ When a sequence is given like $1,1,1,1..$ , it means the coefficients of variables of generating function ,respectively.For example ,

*

*$1,1,1,1,1= 1+x+x^2 +x^3 +x^4 +x^5$


*$1,3,2,2,4,7 = 1+3x+2x^2 +2x^3 +4x^4 +7x^5$


*$0,0,0,0,0,1,1,1=x^5 +x^6+x^7$


*$0,1,0,0,0,4,4,9 =x+4x^5 +4x^6 +9x^7$


*$\binom{5}{0},\binom{5}{1},\binom{5}{2},\binom{5}{3},\binom{5}{4},\binom{5}{5}=1+5x+10x^2+10x^3+5x^4 +x^5 =(1+x)^5$
$1.b-)$ When we comes to "$e^x$ is the egf for counting set" , you must know that when we distribute distinguishable objects into distinguishable boxes , we makes use of exponential generating functions , you can find many example about it in my answers.I am putting here one of them.However ,this is a little long to teach here.I recommend you read this book to begin.
$2-)$ It is given that Schöder path is $\sum_0^\infty( x+xR(x))^k$ , it means that $$( x+xR(x))^0 +( x+xR(x))^1+( x+xR(x))^2+( x+xR(x))^3+( x+xR(x))^4+...$$
Then ,we know that $$\frac{1}{1-x}=x^0 +x^1 +x^2 +x^3 +....$$
So , if we switch $x$ with $x+xR(x)$ , we can reach our aim such that $$\frac{1}{1-(x+xR(x))} =(x+xR(x))^0 +(x+xR(x))^1 +(x+xR(x))^2 +...$$
A: The question about

$e^x$ is the egf for counting "set".

does not have enough context, but I can guess
what it is supposed to be. Given any sequence
$\,a_0,a_1,\dots,\,$ its egf is the formal power
series
$$ A(x) = \text{egf}\,[a_n] := \sum_{n=0}^\infty
a_n\frac{x^n}{n!}. $$
Its gf is the same without $\,n!\,$
in the denominator. Thus, as the first example,
$\,\frac1{1-x}\,$ is the gf for the sequence
$\,1,1,1,\dots\,$ and $\,e^x\,$ is the egf for
the same sequence.
Now suppose we are counting the number of sets
which are the "same up to isomorphism". That is,
there is a
bijection between them. Then, they have the same
number of elements. Therefore, there is only a
single set "up to isomorphism" for each number of
elements. This is the same sequence $\,1,1,1,\dots.$
