Is there an exponential generating function for the sequence $a_n = 2^ {n \choose 2}$? The context is that I want to find the number of simple graphs on n labeled vertices where each connected component has at most 5 vertices. The latter structure I'm okay with but I can't get the generating function for number of simple graphs on n labeled vertices. I have tried a couple of different solutions. First off I defined the generating function as
$A(x) = \sum_{n \geq 0} a_n x^n /n!$
Then letting $b_n = ln(a_n)$ and defining a generic exponential function for B(x).
$B(x) =\sum_{n \geq 0} b_n x^n /n! = \frac{ln(2)}{2} x^2\sum_{n \geq 0} \frac{x^{n-2}} {(n-2)!} = \frac{ln(2)}{2} x^2 e^x $
Then I thought I could write
$A(x) = e^{B(x)} = 2 e^{x^2 e^x/2}$
However this is giving the incorrect sequence.
The other attempt I made was similar but using the power series expansion and getting
$B(x)= ln(2) \frac{x^2}{ (1-x)^3}$
and
$A(x) = 2e^{\frac{x^2}{ (1-x)^3}}$
This is also incorrect.