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%I #29 Feb 21 2024 15:54:27
%S 1,4,51,2460,513619,509709696,2590569730617,71972142178289680,
%T 11572569464349559854105,11332749125368045400133079296,
%U 70775590368575601248957366910425851,2939823814188321813975498471683171002746816,844162736935477006294039214093750952242356035727995,1736712038520659436678773853448507425382701807453031820800000
%N Number of matchings in the n-triangular graph.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentEdgeSet.html">Independent Edge Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JohnsonGraph.html">Johnson Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching.html">Matching</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>
%o (PARI)
%o \\ groups all labeled oriented graphs on n vertices by out degree configuration.
%o OrientedByOutDegrees(n)={ \\ high memory usage and slow for n > 10
%o local(M=Map());
%o my(acc(p,v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
%o my(recurse(p,i,q,v,e)=if(i<0, for(k=0,e,acc(x^k+q, binomial(e,k)*v)), my(t=polcoeff(p,i));for(k=0,t,self()(p,i-1,(t-k+x*k)*x^i+q,binomial(t,k)*v,e+t-k))));
%o my(iterate(v,k,f)=for(i=1,k,v=f(v));v);
%o iterate(Mat([1,1]), n-1, src->M=Map();for(i=1, matsize(src)[1], my(p=src[i,1]); recurse(p,poldegree(p),0,src[i,2],0)); Mat(M))
%o }
%o a(n)={
%o my(v=vector(n\2,n,(2*n)!/(2^n*n!)));
%o my(c(p)=my(h=(poldegree(p)+1)\2); my(r=n-1-sum(i=1,h,polcoeff(p,2*i-1))); (1+sum(i=1,r\2,binomial(r,2*i)*v[i]))*prod(i=1,h,v[i]^(polcoeff(p,2*i)+polcoeff(p,2*i-1))));
%o my(M=OrientedByOutDegrees(n-1));
%o sum(i=1,matsize(M)[1],M[i,2]*c(M[i,1]))
%o } \\ _Andrew Howroyd_, Aug 25 2017
%K nonn
%O 2,2
%A _Eric W. Weisstein_, May 22 2017
%E a(9)-a(12) from _Andrew Howroyd_, Aug 25 2017
%E a(13)-a(14) from _Eric W. Weisstein_, Oct 01 2017
%E a(15) from _Eric W. Weisstein_, Oct 15 2017
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