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Problem 18

Posted 11/01/2015
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This problem is proposed by Ran Pan. If you want to submit your problem, please click here.

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Consider a soccer league consisting of $n$ teams. In a season, each team plays against any other team once. Clearly, there are $\frac{1}{2}(n^2-n)$ games in a season. The rule is that for each match, the winning team gets 3 points and both the teams get 1 point if they tie.

$a(n)$ is the number of all the possible final standings of a soccer league consisting of $n$ teams.

For example, $a(2)=3$ because there are $3$ possible standings for a two-team league: $\{3,0\}$, $\{1,1\}$, $\{0,3\}$.

Find $a(n)$.







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