I've recently come across the Bell numbers, defined as:
\begin{equation*} B_{n+1} = \sum_{k=0}^{n}\binom{n}{k}B_{k}. \end{equation*} The exponential generating function of the Bell numbers is known to be: \begin{equation*} B(x) = e^{e^{x}-1}. \end{equation*}
If I understand this correctly, the x-th Bell number can calso be computed using this generating function, yet when, for example, inserting the value 2 as input, the output is of course not 2; yet $B_{2} = 2$. What exactly does this function imply then---how is this related to Bell numbers?
I realize this might be a very basic question, if there is any relevant literature for this topic I would be grateful.
"Exponential generating function $B(x)$" means when you expand it in a power series $$ e^{e^x-1} = 1+x+{x}^{2}+{\frac {5}{6}}{x}^{3}+{\frac {5}{8}}{x}^{4}+{\frac {13}{ 30}}{x}^{5}+O \left( {x}^{6} \right) $$ then the coefficient of $x^k$ is $B_k/k!$. So if (your example) you want $B_2$, you look at the coefficient of $x^2$, which is $1$, so that $B_2/2!=1$ and therefore $B_2=2! = 2$.
Another example. The coefficient of $x^5$ is $13/30$, so $B_5/5! = 13/30$, from which we get $B_5 = (13/30) 5! = 52$.