A097174
Total number of red nodes among tricolored labeled trees on n nodes.
Original entry on oeis.org
1, 0, 6, 12, 320, 2190, 51492, 685496, 17286768, 348213690, 9956411300, 266065478052, 8737396913544, 287741445880070, 10816320294520860, 420123621828718320, 17913098835916877792, 798053882730994171890, 38192029991097097185108, 1914946396460982552420380
Offset: 1
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Rest[CoefficientList[Series[LambertW[-LambertW[-x]], {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Aug 26 2016 *)
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x='x+O('x^50); Vec(serlaplace(lambertw(-lambertw(-x)))) \\ G. C. Greubel, Nov 15 2017
A097170
Total number of minimal vertex covers among labeled trees on n nodes.
Original entry on oeis.org
1, 2, 3, 40, 185, 3936, 35917, 978160, 14301513, 464105440, 9648558161, 361181788584, 9884595572293, 419174374377136, 14317833123918885, 679698565575210976, 27884513269105178033, 1468696946887669701312
Offset: 1
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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
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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 *)
A097172
Total number of brown nodes among tricolored labeled trees on n nodes.
Original entry on oeis.org
3, 4, 185, 1026, 30457, 362664, 10245825, 195060070, 5907674201, 153676400076, 5199628119985, 169205814335754, 6462995557999905, 249877775352089296, 10749867848389013249, 478345428286978038606, 23013713995857481324969
Offset: 3
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Drop[CoefficientList[Series[-LambertW[-x] - LambertW[-LambertW[-x]]- LambertW[-LambertW[-x]]^2, {x, 0, 20}], x] * Range[0, 20]!, 3] (* Vaclav Kotesovec, Aug 26 2016 *)
A097173
Total number of green nodes among tricolored labeled trees on n nodes.
Original entry on oeis.org
0, 2, 0, 48, 120, 4560, 35700, 1048992, 15514128, 456726240, 10073339100, 323266492560, 9361060088952, 336767513038320, 11913610172869860, 482920107426039360, 19998225191360977440, 909512248720724321472
Offset: 1
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Rest[CoefficientList[Series[LambertW[-LambertW[-x]]^2, {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Aug 26 2016 *)
Showing 1-4 of 4 results.