Positivity and Negativity of Cumulants

Question

There's few examples of distributions whose cumulants can be computed relatively easily. Normal random variables are of course the easiest: the first cumulant is the mean and the second is the variance. All other cumulants vanish. Another relatively simple class of examples are exponential distributions. The cumulants of the $\mbox{Exp}(\lambda)$ distribution are given by $\kappa_j=\lambda^{-j}(j-1)!$. I've recently become interested in the following question. What are examples of distributions with all cumulants positive? Is the property of positive cumulants equivalent to a seemingly unrelated property of the underlying distribution? I am also interested in examples of distributions whose odd cumulants are negative and even cumulants are positive. The only example I can think of so far is when you take a normal with a negative mean.