  Consider a sample of size n drawn from a large, neutral population of haploid
individuals subject to mutation whose genealogy is governed by Kingmans
n-coalescent. Let Sn count the number of segregating sites in this sample under
the infinitely many sites model of Kimura. For fixed sample size n the main
result about Sn is due to Watterson who computed its mean and variance. In our
main result, Theorem 3, we generalize Watterson's result and compute the ith
cumulant of Sn. We find in passing an explicit expression for the cumulants of
the negative binomial distribution in terms of the polylogarithm. This seems to
be the first explicit formula in the literature for the cumulant of arbitrary
order of the negative binomial distribution.
  As an application of this result we obtain straightforward proofs of the Law
of Large Numbers and the Central Limit Theorem for Sn.
