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%I #12 Dec 14 2017 06:04:46
%S 1,1,6,24,320,3270,55482,999656,21718440,544829130,15130478990,
%T 475440344412,16294653237876,613546243029902,25016884214147490,
%U 1100408748640263120,51948228453097163312,2617775548597611727506,140364712844785892810646,7975414423897012183673540
%N Number of maximal matchings among labeled trees on n nodes.
%H S. Coulomb and M. Bauer, <a href="http://arXiv.org/abs/math.CO/0407456">On vertex covers, matchings and random trees</a>
%F Coulomb and Bauer give a g.f.
%p umax := 20 ; u := array(0..umax) ; U := proc() global umax,u ; local resul,n ; resul :=0 ; for n from 0 to umax do resul := resul+u[n]*x^n ; od: end: expU := proc() global umax,u ; taylor(exp(U()),x=0,umax+1) ; end: xexpU := proc() global umax,u ; taylor(x*expU(),x=0,umax+1) ; end: exexpU := proc() global umax,u ; local t ; t := xexpU() ; taylor(exp(-t^2+t+3*U()),x=0,umax+1) ; end: A := expand(taylor(U()-x^2*exexpU(), x=0,umax+1)) ; for n from 0 to umax do u[n] := solve(coeff(A,x,n),u[n]) ; od : F := proc() t := xexpU() ; taylor(-(t+U())^2/2+(1+U()*t)*t+U()-U()^2,x=0,umax+1) ; end: egf := F() ; for n from 1 to umax do n!*coeff(egf,x,n) ; od; # _R. J. Mathar_, Sep 14 2006
%t nmax = 20; egf := -U^2 - (1/2)*(E^U*x + U)^2 + E^U*x*(E^U*U*x + 1) + U;
%t U = 1;
%t Do[U = Normal[x^2*E^(E^(2U)*(-x^2) + E^U*x + 3U) + O[x]^n], {n, 1, nmax}];
%t Rest[Range[0, nmax - 1]!*CoefficientList[egf + O[x]^nmax, x]] (* _Jean-François Alcover_, Dec 14 2017 *)
%Y Cf. A097170, A097172, A097173, A097174, A000169, A000272.
%K nonn
%O 1,3
%A _Ralf Stephan_, Jul 30 2004
%E More terms from _R. J. Mathar_, Sep 14 2006
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