  The score sequence of a tournament is the sequence of the out-degrees of its
vertices arranged in nondecreasing order. The problem of counting score
sequences of a tournament with $n$ vertices is more than 100 years old
(MacMahon 1920). In 2013 Hanna conjectured a surprising and elegant recursion
for these numbers. We settle this conjecture in the affirmative by showing that
it is a corollary to our main theorem, which is a factorization of the
generating function for score sequences with a distinguished index. We also
derive a closed formula and a quadratic time algorithm for counting score
sequences.
