A119743 Triangle read by rows: row n gives number of matchings of size 0<=k<=n (edges) in the complete graph on 2*n >= 2 vertices.

1, 1, 1, 6, 3, 1, 15, 45, 15, 1, 28, 210, 420, 105, 1, 45, 630, 3150, 4725, 945, 1, 66, 1485, 13860, 51975, 62370, 10395, 1, 91, 3003, 45045, 315315, 945945, 945945, 135135, 1, 120, 5460, 120120, 1351350, 7567560, 18918900, 16216200, 2027025, 1, 153, 9180

Offset: 1

Author

Swapnil P. Bhatia (sbhatia(AT)cs.unh.edu), Jul 29 2006

Keywords

• nonn
• tabl

Examples

			For example, T(3,2) is the number of matchings composed of any two edges of the complete graph on 6 vertices. Then T(3,2) = a(3*(3+1)/2+2) = a(8) = 45. Similarly, T(2,2)=a(5)=3 since the only matchings of size 2 on the K_4 are {{0,1},{2,3}}, {{0,3}{1,2}} and {{0,2},{1,3}}.
		

References

• The special case m(n,n) appears in: Flajolet, P. and Noy, M., "Analytic Combinatorics of Chord Diagrams", INRIA Research Report, ISRN INRIA/RR-3914-FR+ENG, March 2000.

Links

• T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras

Programs

Mathematica

Table[(2n)!/((2n-2k)!k! 2^k),{n,10},{k,0,n}]//Flatten (* Harvey P. Dale, Aug 11 2019 *)

Formula

T(n,k)=(2*n)! / ((2*n-2*k)!*k!*2^k).