A097170 Total number of minimal vertex covers among labeled trees on n nodes.

1, 2, 3, 40, 185, 3936, 35917, 978160, 14301513, 464105440, 9648558161, 361181788584, 9884595572293, 419174374377136, 14317833123918885, 679698565575210976, 27884513269105178033, 1468696946887669701312

Offset: 1

Author

Ralf Stephan, Jul 30 2004

Keywords

• nonn

Links

• S. Coulomb and M. Bauer, On vertex covers, matchings and random trees, arXiv:math/0407456 [math.CO], 2004.

Crossrefs

Cf. A097171, A097172, A097173, A097174, A000169, A000272.

Programs

Maple

umax := 20 : u := array(0..umax) : T := proc(z) local resul,n ; global umax,u ; resul :=0 ; for n from 1 to umax do resul := resul +n^(n-1)/n!*z^n ; od : RETURN(taylor(resul,x=0,umax+1)) ; end: 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: xUexpU := proc() global umax,u ; taylor(x*U()*expU(),x=0,umax+1) ; end: exexpU := proc() global umax,u ; taylor(exp(x*expU())-1,x=0,umax+1) ; end: x2e2U := taylor((x*expU())^2,x=0,umax+1) ; A := expand(taylor(xUexpU()-T(x2e2U)*exexpU(), x=0,umax+1)) ; for n from 0 to umax do u[n] := solve(coeff(A,x,n+1),u[n]) ; od ; F := proc() global umax,u ; taylor((1-U())*x*expU()-U()*T(x2e2U)+U()-U()^2/2,x=0,umax+1) ; end: egf := F() ; for n from 0 to umax-1 do n!*coeff(egf,x,n) ; od; # R. J. Mathar, Sep 14 2006

Mathematica

uMax = 20; Clear[u]; u[0] = u[1] = 0; u[2] = 1;
T[x_] := Sum[n^(n - 1)/n!*x^n , {n, 1, uMax}];
U[] = Sum[u[n]*x^n, {n, 0, uMax}];
ExpU[] = Series[Exp[U[]], {x, 0, uMax + 1}];
xUExpU[] = Series[x*U[]*ExpU[], {x, 0, uMax + 1}];
exExpU[] = Series[Exp[x*ExpU[]] - 1, {x, 0, uMax + 1}];
x2e2U = Series[(x*ExpU[])^2, {x, 0, uMax + 1}];
A = Series[xUExpU[] - T[x2e2U]*exExpU[], {x, 0, uMax + 1}] // CoefficientList[#, x]&;
sol = Solve[Thread[A == 0]][[1]];
egf = Series[(1 - U[])*x*ExpU[] - U[]*T[x2e2U] + U[] - U[]^2/2 /. sol, {x, 0, uMax + 1}];
Most[CoefficientList[egf, x]]*Range[0, uMax]! // Rest (* Jean-François Alcover, Dec 11 2017, translated from Maple *)

Formula

Coulomb and Bauer give a g.f.

Extensions

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