A097174 -id:A097174 - cp's OEIS frontend

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

Ralf Stephan, Jul 30 2004

nonn

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

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

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

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 *)

Coulomb and Bauer give a g.f.

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

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

Ralf Stephan, Jul 30 2004

nonn

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

Equals A000169(n)-A097174(n)-A097173(n).

Cf. A097170, A097171, A097173, A097174, A000272.

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 *)

E.g.f.: A(x) = T(x)+T(-T(x))-T(-T(x))^2, with T(x)=Sum[k=1..inf, A000169(k)/k!*x^k].
a(n) = -n^(n-1) * {1 + Sum[l=1..n, (-l/n)^l*(2/l-1)*C(n, l)]}.
a(n) ~ (1-2*LambertW(1)^2)*n^(n-1)/(1+LambertW(1)). - 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

Ralf Stephan, Jul 30 2004

nonn

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

Cf. A097170, A097171, A097172, A097174, A000272.

Rest[CoefficientList[Series[LambertW[-LambertW[-x]]^2, {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Aug 26 2016 *)

E.g.f.: A(x) = T(-T(x))^2, with T(x)=Sum[k=1..inf, A000169(k)/k!*x^k].
a(n) = -2 * n^(n-1) * Sum[l=1..n, (-l/n)^l*(1/l-1)*C(n, l)].
a(n) ~ 2*LambertW(1)^2*n^(n-1)/(1+LambertW(1)). - Vaclav Kotesovec, Aug 26 2016

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

Original entry on oeis.org

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

Offset: 1

Ralf Stephan, Jul 30 2004

nonn

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

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

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

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 *)

Coulomb and Bauer give a g.f.

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

A276231 E.g.f. A(x) satisfies: A(x)^A(x) = LambertW(-x)/(-x).

Original entry on oeis.org

1, 1, 1, 7, 37, 441, 4771, 79213, 1320649, 28318321, 636978151, 16863972621, 475580960317, 15055752973561, 508984025190187, 18802677669334861, 739723172361876241, 31282037176343362785, 1402437758091393319759, 66859536126516402568717, 3362832363918613596662341, 178500985406930615357763241, 9950984335825184802962609491, 582129154096893229447821411597, 35620632904151979409688095495897, 2277073896917989779381561818509201

Offset: 0

Paul D. Hanna, Aug 24 2016

nonn

Let G(x) = (-x)/LambertW(-x), then A(x)^A(x) = 1/G(x) where G(x)^G(x) = 1/exp(x).

a(n) = 0 (mod 3) when n = 6*k+5, k>=0, otherwise a(n) = 1 (mod 3).

			E.g.f.: A(x) = 1 + x + x^2/2! + 7*x^3/3! + 37*x^4/4! + 441*x^5/5! + 4771*x^6/6! + 79213*x^7/7! + 1320649*x^8/8! + 28318321*x^9/9! + 636978151*x^10/10! + 16863972621*x^11/11! + 475580960317*x^12/12! +...
such that A(x)^A(x) = LambertW(-x)/(-x), where
LambertW(-x)/(-x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + 262144*x^7/7! + 4782969*x^8/8! +...+ (n+1)^(n-1)*x^n/n! +...
The logarithm of the e.g.f. A(x) begins
log(A(x)) = x + 6*x^3/3! + 12*x^4/4! + 320*x^5/5! + 2190*x^6/6! + 51492*x^7/7! + 685496*x^8/8! + 17286768*x^9 +...+ A097174(n)*x^n/n! +...
which equals -T(-T(x)), where
T(x) = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 625*x^5/5! + 7776*x^6/6! + 117649*x^7/7! + 2097152*x^8/8! +...+ n^(n-1)*x^n/n! +...

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A097174 (log(A(x))).

CoefficientList[Series[E^LambertW[-LambertW[-x]], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 26 2016 *)

{a(n) = my(A=1+x, W); W=serreverse(x*exp(-x +x^2*O(x^n)))/x;
for(i=0,n, A = W^(1/A) ); n!*polcoeff(A,n)}
for(n=0, 40, print1(a(n), ", "))

E.g.f.: exp( -T(-T(x)) ), where T(x) = Sum_{n>=1} n^(n-1)*x^n/n!.

a(n) ~ n^(n-1)/(1+LambertW(1)). - Vaclav Kotesovec, Aug 26 2016

A115287 Decimal expansion of 1/(1+LambertW(1)).

Original entry on oeis.org

6, 3, 8, 1, 0, 3, 7, 4, 3, 3, 6, 5, 1, 1, 0, 7, 7, 8, 5, 2, 2, 4, 0, 7, 3, 8, 5, 5, 1, 9, 8, 8, 0, 3, 1, 4, 4, 4, 3, 9, 3, 3, 8, 4, 1, 2, 8, 9, 0, 2, 7, 6, 4, 0, 4, 1, 9, 4, 8, 3, 1, 9, 3, 6, 5, 0, 3, 4, 2, 1, 0, 1, 0, 5, 6, 7, 6, 0, 0, 8, 3, 0, 4, 1, 0, 0, 1, 8, 5, 2, 5, 1, 0, 5, 2, 7, 4, 8, 3, 3, 1, 5, 7, 0, 9

Offset: 0

Eric W. Weisstein, Jan 19 2006

			0.63810374336511077852...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Cristina B. Corcino, Roberto B. Corcino and István Mező, Continued fraction expansions for the Lambert W function, Aequationes mathematicae, Vol. 93, No. 2 (2019), pp. 485-49.
Mathematics Stack Exchange, Interesting integral related to the Omega Constant/Lambert W Function, 2011.
Victor H. Moll, Some Questions in the Evaluation of Definite Integrals, MAA Short Course, San Antonio, TX, January 2006.
Eric Weisstein's World of Mathematics, Omega Constant.
Eric Weisstein's World of Mathematics, Definite Integral.

Cf. A006153, A030178, A097172, A097173, A097174, A265953, A276231, A302399.

RealDigits[1/(1 + ProductLog[1]), 10, 105] // First (* Jean-François Alcover, Feb 07 2013 *)

1/(1 + lambertw(1)) \\ G. C. Greubel, Nov 05 2017

Equals Integral_{x=-oo..oo} 1/(Pi^2 + (exp(x)-x)^2) dx (discovered by Victor Adamchik). - Amiram Eldar, Jul 04 2021

A340473 a(n) = n! [x^n] W(-W(x))/(-W(x)), where W(x) is the Lambert W function.

Original entry on oeis.org

1, 1, 1, 7, 13, 321, 31, 42673, -214983, 12251809, -156239909, 6366130761, -135725103227, 5265915854785, -155145910919817, 6318044844152161, -232403136941014799, 10299509100942804033, -446889500139353805773, 21789892230658085847673, -1078684347590588362463619

Offset: 0

Peter Luschny, Jan 08 2021

sign

Let LW(x) = W(-W(x))/(-W(x)) denote the function in the definition and let T(x) = -W(-x) be Euler's tree function A000169, and L(x) = W(-x)/(-x) the labeled tree function A000272, then LW(x) = L(W(x)), and TW(x) := -T(W(-x)) is A097174, and RW(x) := T(-W(-x)) is A207833.

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A097174, A177885, A207833, A227176, A340474.

Cf. A000169, A000272.

W := x -> LambertW(x): gf := W(-W(x))/(-W(x)):
ser := series(gf, x, 24): seq(n!*coeff(ser, x, n), n=0..20);

gf := -ProductLog[-ProductLog[x]]/ProductLog[x];
Range[0, 20]! CoefficientList[Series[gf, {x, 0, 20}], x]

my(x='x+O('x^25)); Vec(serlaplace(lambertw(-lambertw(x))/(-lambertw(x)))) \\ Michel Marcus, Jan 09 2021

A340474 a(n) = n! [x^n] LW(T(x)), where T(x) = -W(-x) Euler's tree function, W(x) is the Lambert W function, and LW(x) = W(-W(x))/(-W(x)) (A340473).

Original entry on oeis.org

1, 1, 3, 22, 209, 2756, 43717, 839686, 18581425, 470707192, 13352676101, 420875581754, 14566375690297, 549877190829604, 22472783629465093, 989043215802778966, 46631075599107558113, 2345376059569552767344, 125350843842721213505029, 7095169059445749303612946

Offset: 0

Peter Luschny, Jan 09 2021

nonn

Cf. A340473, A097174, A177885, A207833, A227176.

Cf. A000169, A000272.

W := x -> LambertW(x): T := x -> -W(-x): LW := x -> W(-W(x))/(-W(x)):
ser := series(LW(T(x)), x, 24): seq(n!*coeff(ser, x, n), n=0..19);

cp's OEIS Frontend

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

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A097172 Total number of brown nodes among tricolored labeled trees on n nodes.

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A097173 Total number of green nodes among tricolored labeled trees on n nodes.

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

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A276231 E.g.f. A(x) satisfies: A(x)^A(x) = LambertW(-x)/(-x).

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A115287 Decimal expansion of 1/(1+LambertW(1)).

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A340473 a(n) = n! [x^n] W(-W(x))/(-W(x)), where W(x) is the Lambert W function.

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A340474 a(n) = n! [x^n] LW(T(x)), where T(x) = -W(-x) Euler's tree function, W(x) is the Lambert W function, and LW(x) = W(-W(x))/(-W(x)) (A340473).

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