%I #41 Jan 16 2023 17:53:47

%S 3,45,210,630,1485,3003,5460,9180,14535,21945,31878,44850,61425,82215,

%T 107880,139128,176715,221445,274170,335790,407253,489555,583740,

%U 690900,812175,948753,1101870,1272810,1462905,1673535,1906128,2162160,2443155

%N Consider 2n tennis players; a(n) is the number of matches needed to let every possible pair play each other.

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

%H Vincenzo Librandi, <a href="/A062346/b062346.txt">Table of n, a(n) for n = 2..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = n*(4*n^3 - 12*n^2 + 11*n - 3)/2. - Swapnil P. Bhatia (sbhatia(AT)cs.unh.edu), Jul 20 2006

%F a(n+1) = (2*n+2)*(2*n+1)*(2*n)*(2*n-1)/8. - _James Mahoney_, Oct 19 2011

%F G.f.: 3*x^2*(1 + 10*x + 5*x^2)/(1 - x)^5. - _Vincenzo Librandi_, Oct 13 2013

%F a(n) = binomial(2*n^2-3*n+1, 2). - _Wesley Ivan Hurt_, Oct 14 2013

%F a(n) = A014105(n-1)*(A014105(n-1)-1)/2. - _Bruno Berselli_, Dec 28 2016

%e a(2)=3: given players a,b,c,d, the matches needed are (ab,cd), (ac,bd), (ad,bc).

%e 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}}.

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

%t CoefficientList[Series[3 (1 + 10 x + 5 x^2)/(1 - x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 13 2013 *)

%t LinearRecurrence[{5,-10,10,-5,1},{3,45,210,630,1485},40] (* _Harvey P. Dale_, Nov 22 2022 *)

%o (PARI) a(n) = n*(n-1)*(2*n-3)*(2*n-1)/2; \\ _Joerg Arndt_, Oct 13 2013

%o (Magma) [n*(n-1)*(2*n-3)*(2*n-1)/2: n in [2..40]]; // _Vincenzo Librandi_, Oct 13 2013

%Y Cf. A014105.

%K nonn,easy

%O 2,1

%A _Michel ten Voorde_, Jul 06 2001

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