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A062346
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Consider 2n tennis players; a(n) is the number of matches needed to let every possible pair play each other.
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1
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3, 45, 210, 630, 1485, 3003, 5460, 9180, 14535, 21945, 31878, 44850, 61425, 82215, 107880, 139128, 176715, 221445, 274170, 335790, 407253, 489555, 583740, 690900, 812175, 948753, 1101870, 1272810, 1462905, 1673535, 1906128, 2162160, 2443155
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OFFSET
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2,1
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COMMENTS
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Number of matchings of size two (edges) in a complete graph on 2n vertices.
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LINKS
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FORMULA
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a(n) = n*(4*n^3 - 12*n^2 + 11*n - 3)/2. - Swapnil P. Bhatia (sbhatia(AT)cs.unh.edu), Jul 20 2006
a(n+1) = (2*n+2)*(2*n+1)*(2*n)*(2*n-1)/8. - James Mahoney, Oct 19 2011
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EXAMPLE
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a(2)=3: given players a,b,c,d, the matches needed are (ab,cd), (ac,bd), (ad,bc).
For example, for the K_4 on vertices {0,1,2,3} the possible matchings of size two are: {{0,1}, {2,3}}, {{0,3},{1,2}} and {{0,2},{1,3}}.
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MAPLE
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MATHEMATICA
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CoefficientList[Series[3 (1 + 10 x + 5 x^2)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 13 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {3, 45, 210, 630, 1485}, 40] (* Harvey P. Dale, Nov 22 2022 *)
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PROG
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(PARI) a(n) = n*(n-1)*(2*n-3)*(2*n-1)/2; \\ Joerg Arndt, Oct 13 2013
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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More terms from Swapnil P. Bhatia (sbhatia(AT)cs.unh.edu), Jul 20 2006
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STATUS
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approved
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