Counting tournament score sequences

The score sequence of a tournament is the out-degrees its vertices arranged in nondecreasing order. problem counting sequences with n n more than 100 years old [Quart. J. Math. 49 (1920), pp. 1–36]. In 2013 Hanna conjectured surprising and elegant recursion for these numbers. We settle this conjecture affirmative by showing that it corollary to our main theorem, which factorization generating function distinguished index. also derive closed formula quadratic time algorithm sequences.