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A062346 Consider 2n tennis players; a(n) is the number of matches needed to let every possible pair play each other. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

Number of matchings of size two (edges) in a complete graph on 2n vertices.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 2..1000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

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

G.f.: 3*x^2*(1 + 10*x + 5*x^2)/(1 - x)^5. - Vincenzo Librandi, Oct 13 2013

a(n) = binomial(2*n^2-3*n+1, 2). - Wesley Ivan Hurt, Oct 14 2013

a(n) = A014105(n-1)*(A014105(n-1)-1)/2. - Bruno Berselli, Dec 28 2016

EXAMPLE

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}}.

MAPLE

A062346:=n->n*(n-1)*(2*n-3)*(2*n-1)/2; seq(A062346(k), k=2..100); # Wesley Ivan Hurt, Oct 14 2013

MATHEMATICA

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 *)

PROG

(PARI) a(n) = n*(n-1)*(2*n-3)*(2*n-1)/2; \\ Joerg Arndt, Oct 13 2013

(Magma) [n*(n-1)*(2*n-3)*(2*n-1)/2: n in [2..40]]; // Vincenzo Librandi, Oct 13 2013

CROSSREFS

Cf. A014105.

Sequence in context: A062270 A069955 A289193 * A002682 A073595 A117972

Adjacent sequences: A062343 A062344 A062345 * A062347 A062348 A062349

KEYWORD

nonn,easy,changed

AUTHOR

Michel ten Voorde Jul 06 2001

EXTENSIONS

More terms from Swapnil P. Bhatia (sbhatia(AT)cs.unh.edu), Jul 20 2006

STATUS

approved

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Last modified December 3 09:50 EST 2022. Contains 358517 sequences. (Running on oeis4.)