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%I #6 Aug 11 2019 16:40:58
%S 1,1,1,6,3,1,15,45,15,1,28,210,420,105,1,45,630,3150,4725,945,1,66,
%T 1485,13860,51975,62370,10395,1,91,3003,45045,315315,945945,945945,
%U 135135,1,120,5460,120120,1351350,7567560,18918900,16216200,2027025,1,153,9180
%N 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.
%D 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.
%H T. Copeland, <a href="http://tcjpn.wordpress.com/2012/11/29/infinigens-the-pascal-pyramid-and-the-witt-and-virasoro-algebras/">Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras</a>
%F T(n,k)=(2*n)! / ((2*n-2*k)!*k!*2^k).
%e 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}}.
%t Table[(2n)!/((2n-2k)!k! 2^k),{n,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Aug 11 2019 *)
%K nonn,tabl
%O 1,4
%A Swapnil P. Bhatia (sbhatia(AT)cs.unh.edu), Jul 29 2006
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