Number of regular tournaments

Question

A regular tournament is a tournament where each player has the same number of wins. Since each player plays $n-1$ games, a regular tournament must have an odd number of players. My question is - 'how many nonisomorphic regular tournaments are possible on $n=2k-1$ vertices?'

Answer 1

This is the sequence OEIS A007079, save that $a_n$ is defined there to be the number of labelled regular tournaments on $2n+1$ nodes (rather than $2n-1$). (I’m assuming that you want the players to be individually identifiable, so that you’re interested in labelled tournaments; if not, you want OEIS A096368.)

The OEIS entry has very little information; it does give a formula,

$$a_n=\left[(x_1x_2\ldots x_n)^{(n-1)/2}\right]\prod_{1\le j<k\le n}(x_j+x_k)\;,$$

where the square brackets are the ‘coefficient of ... in ...’ operator.