It is very easy to have an ANN that can compute the average of an input vector, but is it possible to have an ANN that can generate the output vector given just an average of the output vector as the input?
No, if the numbers do not follow a pattern other than "having a mean of X".
If the information available is only mean, consider the following example.
"A set of numbers has mean 10."
I can return ANY set of two numbers that averages to 10.
In fact I can return a set of ANY length as long as it averages to 10.
If the only known information is the mean, you just end up with basically infinite possible solutions.
If there are other information such as standard deviation or range, you can narrow down the possible solutions from "infinitely many" to "so many". But then, why would you do this?