A062346 Consider 2n tennis players; a(n) is the number of matches needed to let every possible pair play each other.

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

Offset: 2

Author

Michel ten Voorde, Jul 06 2001

Keywords

• nonn
• easy

Comments

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

Examples

			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}}.
		

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).

Crossrefs

Cf. A014105.

Programs

Magma

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

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 *)

PARI

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

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

Extensions

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