Problem 849

The Tournament

In a tournament there are $n$ teams and each team plays each other team twice. A team gets two points for a win, one point for a draw and no points for a loss.

With two teams there are three possible outcomes for the total points. $(4, 0)$ where a team wins twice, $(3, 1)$ where a team wins and draws, and $(2, 2)$ where either there are two draws or a team wins one game and loses the other. Here we do not distinguish the teams and so $(3, 1)$ and $(1, 3)$ are considered identical.

Let $F (n)$ be the total number of possible final outcomes with $n$ teams, so that $F (2) = 3$ .

You are also given $F (7) = 32923$ .

Find $F (100)$ . Give your answer modulo $10^{9} + 7$ .

锦标赛

在一场锦标赛中，共有 $n$ 支队伍参赛，每支队伍需与任意其它队伍各对战两次。获胜的队伍得两分，平局得一分，落败不得分。

只有两支队伍时，共有三种可能的得分结果。若一支队伍两次都获胜则结果是 $(4, 0)$ ，若一支队伍赢一次平一次则结果是 $(3, 1)$ ，若两次都是平局或者各自一胜一负则结果是 $(2, 2)$ 。得分结果不区分队伍，因此 $(3, 1)$ 和 $(1, 3)$ 视为相同的结果。

记 $F (n)$ 为有 $n$ 支队伍时所有可能的最终得分结果的数目，因此 $F (2) = 3$ 。

已知 $F (7) = 32923$ 。

求 $F (100)$ ，并将你的答案对 $10^{9} + 7$ 取余。