It is well known that a probability distribution whose cumulants are all zero except for the first two is gaussian:
$$ f(x,\mu,\sigma^2) \propto\exp(-\frac{(x-\mu)^2}{2\sigma^2}) $$
So I'm wondering what the equivalent formula is for a distribution with three nonzero cumulants, or better yet a formula for n cumulants if that exists somewhere.