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
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
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
KEYWORD
nonn,easy
AUTHOR
Michel ten Voorde, Jul 06 2001
EXTENSIONS
More terms from Swapnil P. Bhatia (sbhatia(AT)cs.unh.edu), Jul 20 2006
STATUS
approved