A097171 Number of maximal matchings among labeled trees on n nodes.

1, 1, 6, 24, 320, 3270, 55482, 999656, 21718440, 544829130, 15130478990, 475440344412, 16294653237876, 613546243029902, 25016884214147490, 1100408748640263120, 51948228453097163312, 2617775548597611727506, 140364712844785892810646, 7975414423897012183673540

Offset: 1

Author

Ralf Stephan, Jul 30 2004

Keywords

• nonn

Links

• S. Coulomb and M. Bauer, On vertex covers, matchings and random trees

Crossrefs

Cf. A097170, A097172, A097173, A097174, A000169, A000272.

Programs

Maple

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

Mathematica

nmax = 20; egf := -U^2 - (1/2)*(E^U*x + U)^2 + E^U*x*(E^U*U*x + 1) + U;
U = 1;
Do[U = Normal[x^2*E^(E^(2U)*(-x^2) + E^U*x + 3U) + O[x]^n], {n, 1, nmax}];
Rest[Range[0, nmax - 1]!*CoefficientList[egf + O[x]^nmax, x]] (* Jean-François Alcover, Dec 14 2017 *)

Formula

Coulomb and Bauer give a g.f.

Extensions

More terms from R. J. Mathar, Sep 14 2006