A038049 -id:A038049 - cp's OEIS frontend

A007889 Number of intransitive (or alternating, or Stanley) trees: vertices are [0,n] and for no i

1, 1, 2, 7, 36, 246, 2104, 21652, 260720, 3598120, 56010096, 971055240, 18558391936, 387665694976, 8787898861568, 214868401724416, 5636819806209792, 157935254554567296, 4707152127520549120, 148704074888134683520, 4963548160096887021056, 174553183413968718996736

Offset: 0

Alexander Postnikov [ apost(AT)math.mit.edu ]

Number of local binary search trees (i.e. labeled binary trees such that every left child has a smaller label than its parent and every right child has a larger label than its parent) on n vertices. Example: a(3)=7 because we have 3L2L1, 2L1R3, 3L1R2, 1R2R3, 1R3L2, 2R3L1 (Li means left child labeled i, RI means right child labeled i) and root 2 with left child 1 and right child 3. - Emeric Deutsch, Nov 24 2004

Number of regions of the Linial arrangement. - Ira M. Gessel, Nov 01 2023

I. M. Gelfand, M. I. Graev and A. Postnikov, Combinatorics of hypergeometric functions associated with positive roots, in Arnold-Gelfand Mathematical Seminars: Geometry and Singularity Theory, Birkhäuser, 1997.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.41(a).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
C. Chauve, S. Dulucq and A. Rechnitzer, Enumerating alternating trees, J. Combin. Theory Ser. A 94 (2001), 142-151.
Sergey Fomin and Grigory Mikhalkin, Labeled floor diagrams for plane curves, arXiv:0906.3828, 2009-2010. [_N. J. A. Sloane_, Sep 27 2010]
G. Hetyei, Efron's coins and the Linial arrangement, arXiv preprint arXiv:1511.04482 [math.CO], 2015.
D. E. Knuth, Letter to Daniel Ullman and others, Apr 29 1997 [Annotated scanned copy, with permission]
A. Postnikov, Intransitive Trees, J. Combin. Theory Ser. A 79 (1997), 360-366.
Richard P. Stanley, Hyperplane arrangements, interval orders, and trees, Proc. Natl. Acad. Sci. USA 93 (1996), 2620-2625.
Index entries for sequences related to trees

Cf. A038049, A138860, A283828, A323841.

Row sums of A029847.

f:= n->1/(2^n*(n+1))*add(binomial(n+1, k)*k^n, k=1..(n+1)): seq(f(n), n=0..19);

With[{nn=20},CoefficientList[Series[-2/x LambertW[-1/2x Exp[x/2]], {x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Aug 12 2011 *)
Table[1/((n+1)2^n) Sum[Binomial[n+1,k]k^n,{k,n+1}],{n,0,20}] (* Harvey P. Dale, Apr 21 2012 *)

{a(n)=local(A=1+x);for(i=0,n,A=exp(x*(1+A)/2 +x*O(x^n)));n!*polcoeff(A,n)} \\ Paul D. Hanna, Mar 29 2008

/* Coefficients of A(x)^p are given by: */ {a(n,p=1)=(1/2^n)*sum(k=0,n,binomial(n,k)*p*(k+p)^(n-1))} \\ Vladeta Jovovic and Paul D. Hanna, Apr 03 2008

def A007889(n) : return add(binomial(n,k)*(k+1)^(n-1) for k in (0..n))/2^n
for n in (0..19) : print(A007889(n)) # Peter Luschny, Feb 29 2012

a(n) = (1/((n+1)*2^n))*Sum_{k=1..n+1} C(n+1,k)*k^n.
E.g.f. A(x) satisfies: A(x) = exp( x*(1 + A(x))/2 ). E.g.f. A(x) equals the inverse function of 2*log(x)/(1+x). - Paul D. Hanna, Mar 29 2008
E.g.f.: -2/x*LambertW(-1/2*x*exp(1/2*x)). - Vladeta Jovovic, Mar 29 2008
From Vladeta Jovovic and Paul D. Hanna, Apr 03 2008: (Start)
Powers of e.g.f.: If A(x)^p = Sum_{n>=0} a(n,p)*x^n/n! then a(n,p) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*p*(k+p)^(n-1).
Let A(x) = e.g.f. of A007889, B(x) = e.g.f. of A138860 where B(x) = exp( x*[B(x) + B(x)^2]/2 ); then B(x) = A(x*B(x)) = (1/x)*Series_Reversion(x/A(x)) and A(x) = B(x/A(x)) = x/Series_Reversion(x*B(x)). (End)
For n>=2, a(n)=Sum_{1,...,floor(n/2)}binomial(n-1, 2k-1)*k^(n-2). [Vladimir Shevelev, Mar 21 2010]
For n>0, a(n) = A088789(n+1)*2/(n+1). [Vaclav Kotesovec, Dec 26 2011]

A216857 Number of connected functions from {1,2,...,n} into a subset of {1,2,...,n} that have a fixed point summed over all subsets.

Original entry on oeis.org

0, 1, 4, 24, 224, 2880, 47232, 942592, 22171648, 600698880, 18422374400, 630897721344, 23864653578240, 988197253808128, 44460603225407488, 2159714024218951680, 112652924603290615808, 6280048587936003784704, 372616014329572403183616, 23445082059018189741752320, 1559275240299007139066675200

Offset: 0

Geoffrey Critzer, Sep 17 2012

nonn

Essentially the same as A038049.
Also the number of rooted trees whose nodes are labeled with the blocks of a set partition of {1,2,...,n} each having a distinguished element. (See A000248.)
The bijection is straightforward. The trees correspond to functional digraphs mapping the distinguished elements towards the root. All the elements within each block are mapped to the distinguished element of that block. The distinguished element in the root node is the fixed point.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.

Cf. A048802, A072034.

With[{nmax = 20}, CoefficientList[Series[-LambertW[-x*Exp[x]], {x, 0, nmax}], x]*Range[0, nmax]!] (* modified by G. C. Greubel, Nov 15 2017 *)

for(n=0,30, print1(sum(k=1,n, binomial(n,k)*k^(n-1)), ", ")) \\ G. C. Greubel, Nov 15 2017

my(x='x+O('x^50)); concat([0], Vec(serlaplace(-lambertw(-x*exp(x))))) \\ G. C. Greubel, Nov 15 2017

E.g.f.: T(x*exp(x)) where T(x) is the e.g.f. for A000169.
a(n) = Sum_{k=1..n} binomial(n,k)*k^(n-1).
a(n) ~ sqrt(1+LambertW(exp(-1))) * n^(n-1) / (exp(n)*LambertW(exp(-1))^n). - Vaclav Kotesovec, Jul 09 2013
O.g.f.: Sum_{n>=0} n^(n-1)* x^n / (1 - n*x)^(n+1). - Paul D. Hanna, May 22 2018
E.g.f.: the compositional inverse of A(x) is -A(-x). - Alexander Burstein, Aug 11 2018

A349562 Number of labeled rooted forests with 2-colored leaves.

Original entry on oeis.org

1, 2, 8, 56, 576, 7872, 134656, 2771456, 66744320, 1842237440, 57354338304, 1988721131520, 76015173369856, 3175757373243392, 143980934947930112, 7040807787705663488, 369414622819764928512, 20700889684976244621312, 1233951687316746828513280, 77963762014950356953333760

Offset: 0

Alexander Burstein, Nov 22 2021

nonn

a(n) is the number of labeled trees on vertices 0,1,...,n rooted at 0, where all leaves have 2 colors (except the singleton tree 0 has only 1 color).

			a(2)=8 counts trees 0-1-2B, 0-1-2R, 0-2-1B, 0-2-1R, 1B-0-2B, 1B-0-2R, 1R-0-2B, 1R-0-2R (where B and R stand for colors Blue and Red).

Seiichi Manyama, Table of n, a(n) for n = 0..372
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A000007, A007889, A216857, A038049, A201595,

CoefficientList[u/.AsymptoticSolve[u-E^(x(1+u))==0,u->1,{x,0,24}][[1]],x]Factorial/@Range[0,24]
nmax = 20; CoefficientList[Series[-LambertW[-x*Exp[x]]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 25 2021 *)

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k+1)^(k-1)*x^k/(1-(k+1)*x)^(k+1))) \\ Seiichi Manyama, Nov 26 2021

a(n) = Sum_{k=0..n} binomial(n,k)*(k+1)^(n-1).
a(n) = A216857(n+1)/(n+1).
a(n) = A038049(n+1)/(n+1) for n>=1, and a(0) = A038049(1)/2.
a(n) = 2*A201595(n) - A000007(n).
E.g.f. satisfies: A(x) = e^(x*(1 + A(x))).
E.g.f. satisfies: A(-x*A(x)) = 1/A(x).
From Vaclav Kotesovec, Nov 25 2021: (Start)
E.g.f.: -LambertW(-x*exp(x))/x.
a(n) ~ sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (exp(n) * LambertW(exp(-1))^(n+1)).
(End)
From Seiichi Manyama, Nov 26 2021: (Start)
G.f.: Sum_{k>=0} (k+1)^(k-1) * x^k/(1 - (k+1)*x)^(k+1).
a(n) = 2^n * A007889(n). (End)

A029856 Number of rooted trees with 2-colored leaves.

Original entry on oeis.org

2, 2, 5, 13, 37, 108, 332, 1042, 3360, 11019, 36722, 123875, 422449, 1453553, 5040816, 17599468, 61814275, 218252584, 774226549, 2758043727, 9862357697, 35387662266, 127374191687, 459783039109, 1664042970924, 6037070913558, 21951214425140, 79981665585029

Offset: 1

Christian G. Bower

Alois P. Heinz, Table of n, a(n) for n = 1..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 768
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Essentially the same as A036249.

Cf. A000081, A029857, A038049.

A:= proc(n) option remember; if n=0 then 0 else convert(series(x+x* exp(sum(subs(x=x^i, A(n-1))/i, i=1..n-1)), x=0, n+1), polynom) fi end; a:= n-> coeff(A(n), x,n): seq(a(n), n=1..25); # Alois P. Heinz, Aug 22 2008
# second Maple program:
with(numtheory): a:= proc(n) option remember; local d,j; if n<=1 then 2*n else (add(d*a(d), d=divisors(n-1)) +add(add(d*a(d), d=divisors(j)) *a(n-j), j=1..n-2))/ (n-1) fi end: seq(a(n), n=1..25); # Alois P. Heinz, Sep 06 2008

a[n_] := a[n] = If [n <= 1, 2*n, (Sum[d*a[d], {d, Divisors[n-1]}] + Sum[Sum[d*a[d], {d, Divisors[j]}]*a[n-j], {j, 1, n-2}])/(n-1)]; Array[a, 25] (* Jean-François Alcover, Mar 13 2015, after Alois P. Heinz *)

{a(n)=local(A=x+x*O(x^n));for(i=1,n, A=x+x*exp(sum(m=1,n,subst(A,x,x^m)/m)));polcoeff(A,n,x)} \\ Paul D. Hanna, Oct 19 2005

Shifts left under Euler transform.
G.f. satisfies: A(x) = x + x*exp( Sum_{n>=1} A(x^n)/n ). - Paul D. Hanna, Oct 19 2005
a(n) ~ c * d^n / n^(3/2), where d = 3.848442876944251389076286931217197... and c = 0.48335853985605895591573724406549734... - Vaclav Kotesovec, Mar 29 2014

A088789 E.g.f.: REVERT(2x/(1+exp(x))) = Sum_{n>=0} a(n)x^n/n!.

Original entry on oeis.org

0, 1, 1, 3, 14, 90, 738, 7364, 86608, 1173240, 17990600, 308055528, 5826331440, 120629547584, 2713659864832, 65909241461760, 1718947213795328, 47912968352783232, 1421417290991105664, 44717945211445216640, 1487040748881346835200, 52117255681017313721088

Offset: 0

Paul D. Hanna, Oct 15 2003

a(n+1) is also number of ways to place n nonattacking composite pieces semi-rook + semi-bishop on an n X n board. Two semi-bishops (see A187235) do not attack each other if they are in the same northwest-southeast diagonal. Two semi-rooks do not attack each other if they are in the same column (see also semi-queens, A099152). - Vaclav Kotesovec, Dec 22 2011

Alois P. Heinz, Table of n, a(n) for n = 0..200
V. Dotsenko, Pattern avoidance in labelled trees, arXiv preprint arXiv:1110.0844 [math.CO], 2011.
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 716-719.
R. W. Whitty, Rook polynomials on two-dimensional surfaces and graceful labellings of graphs, Discrete Math., 308 (2008), 674-683.

Main diagonal of A378561 (shifted).

a:= n->coeff(series(x/2-LambertW(-1/2*x*exp(1/2*x)), x=0, n+1), x, n)*n!:
seq(a(n), n=0..30); # Alois P. Heinz, Aug 14 2008

Table[n!/2^n*Sum[2^j/j!*StirlingS2[n-1,n-j],{j,1,n}],{n,1,50}] (* Vaclav Kotesovec, Dec 25 2011 *)
With[{nmax = 50}, CoefficientList[Series[x/2 - LambertW[-x*Exp[x/2]/2], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 14 2017 *)

a(n)=local(A); if(n<0,0,A=x+O(x^n);n!*polcoeff(serreverse(2*x/(1 + exp(x))), n))

x='x+O('x^50); concat([0], Vec(serlaplace(x/2 - lambertw(-x*exp(x/2)/2)))) \\ G. C. Greubel, Nov 14 2017

E.g.f.: x/2 - LambertW(-x*exp(x/2)/2). - Vladeta Jovovic, Feb 12 2008
a(n) = (1/2^n)*Sum_{k=1..n} binomial(n,k)*k^(n-1) = A038049(n)/2^n, n>1. - Vladeta Jovovic, Feb 12 2008
Asymptotics: a(n)/(n-2)! ~ b * q^(n-1) * sqrt(n), where q = 1/(2*LambertW(1/exp(1))) = 1.795560738334311... is the root of the equation 2*q = exp(1+1/(2*q)) and b = 1/(2*LambertW(1/exp(1))) * sqrt((1+LambertW(1/exp(1)))/(2*Pi)) = 0.8099431005... - Vaclav Kotesovec, Dec 22 2011, updated Sep 25 2012

More terms from Alois P. Heinz, Aug 14 2008

Minor edits by Vaclav Kotesovec, Mar 31 2014

A214225 E.g.f. satisfies: A(x) = x/(1 - tanh(A(x))).

Original entry on oeis.org

1, 2, 12, 112, 1440, 23616, 471296, 11085824, 300349440, 9211187200, 315448860672, 11932326789120, 494098626904064, 22230301612703744, 1079857012109475840, 56326462301645307904, 3140024293968001892352, 186308007164786201591808, 11722541029509094870876160

Offset: 1

Paul D. Hanna, Jul 07 2012

nonn

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! +...
Related expansions:
A(x) = x + x*tanh(x) + d/dx x^2*tanh(x)^2/2! + d^2/dx^2 x^3*tanh(x)^3/3! + d^3/dx^3 x^4*tanh(x)^4/4! +...
log(A(x)/x) = tanh(x) + d/dx x*tanh(x)^2/2! + d^2/dx^2 x^2*tanh(x)^3/3! + d^3/dx^3 x^3*tanh(x)^4/4! +...
A(x)/x = 1 + x + 4*x^2/2! + 28*x^3/3! + 288*x^4/4! + 3936*x^5/5! + 67328*x^6/6! +...+ A201595(n)*x^n/n! +...
tanh(A(x)) = x + 2*x^2/2! + 10*x^3/3! + 88*x^4/4! + 1096*x^5/5! + 17616*x^6/6! + 346704*x^7/7! + 8072576*x^8/8! +...

G. C. Greubel, Table of n, a(n) for n = 1..370

Cf. A201595, A202617, A214222, A214223, A214224.

Rest[CoefficientList[InverseSeries[Series[x-x*Tanh[x],{x,0,20}],x],x]*Range[0,20]!] (* Vaclav Kotesovec, Sep 17 2013 *)
Flatten[{1,Table[1/2*Sum[Binomial[n,k]*k^(n-1),{k,0,n}],{n,2,20}]}] (* Vaclav Kotesovec, Sep 17 2013 *)

{a(n)=(1/2)*sum(k=0,n,binomial(n,k)*k^(n-1))}
for(n=1, 25, print1(a(n), ", "))

{a(n)=(n-1)!*polcoeff(x/(1 - tanh(x+x*O(x^n)))^n,n)}

{a(n)=n!*polcoeff(serreverse(x-x*tanh(x+x*O(x^n))), n)}

{a(n)=n!*polcoeff(sum(k=1, n, k^(k-1)*cosh(k*x +x*O(x^n))*x^k/k! +x*O(x^n)), n)} \\ Paul D. Hanna, Nov 20 2012
for(n=1, 25, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^m*tanh(x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*tanh(x+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}

E.g.f. A(x) satisfies:
(1) A(x - x*tanh(x)) = x.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*tanh(x)^n/n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*tanh(x)^n/n! ).
(4) A(x) = Sum_{n>=1} n^(n-1) * cosh(n*x) * x^n / n!. - Paul D. Hanna, Nov 20 2012
(5) A(x) = log(G(x)) where G(x) = exp(x*(1+G(x)^2)/2) is the e.g.f. of A202617. - Paul D. Hanna, Nov 20 2012
a(n) = n*A201595(n-1).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*k^(n-1).
a(n) = (n-1)! * [x^n] x/(1 - tanh(x))^n.
a(n) = A038049(n)/2. - R. J. Mathar, Peter Bala, Mar 24 2013
a(n) ~ 1/2 * n^(n-1) * sqrt((1+LambertW(1/exp(1)))) / (exp(1)*LambertW(1/exp(1)))^n. - Vaclav Kotesovec, Sep 17 2013

A100526 Number of local binary search trees (i.e., labeled binary trees such that every left child has a smaller label than its parent and every right child has a larger label than its parent) with n vertices such that the root has only one child.

Original entry on oeis.org

1, 2, 6, 28, 180, 1476, 14728, 173216, 2346480, 35981200, 616111056, 11652662880, 241259095168, 5427319729664, 131818482923520, 3437894427590656, 95825936705566464, 2842834581982211328, 89435890422890433280, 2974081497762693670400, 104234511362034627442176

Offset: 1

Emeric Deutsch, Nov 24 2004

nonn

			a(3)=6 because we have 3L2L1, 2L1R3, 3L1R2, 1R2R3, 1R3L2, 2R3L1 (Li means left child labeled i, RI means right child labeled i).
E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 28*x^4/4! + 180*x^5/5! + 1476*x^6/6! +...
Given G(x) such that G( sqrt( G(x^2*exp(-x)) ) ) = x, where
G(x) = x + 1/2*x^2 + 1/8*x^3 + 1/12*x^4 + 77/384*x^5 + 23/120*x^6 + 2077/46080*x^7 + 179/5040*x^8 + 239525/2064384*x^9 +...+ A273952(n)*x^n/(2^(n-1)*(n-1)!) +...
then A(x) = G( sqrt( G(x^2*exp(x)) ) ). - _Paul D. Hanna_, Jun 06 2016

G. C. Greubel, Table of n, a(n) for n = 1..400
C. Chauve, S. Dulucq and A. Rechnitzer, Enumerating alternating trees, J. Combin. Theory Ser. A 94 (2001), 142-151.
A. Postnikov, Intransitive Trees, J. Combin. Theory Ser. A 79 (1997), 360-366.

Cf. A007889, A273952.

[2^(1-n)*(&+[ k^(n-1)*Binomial(n, k): k in [1..n]]): n in [1..40]]; // G. C. Greubel, Mar 27 2023

seq((1/2^(n-1))*add(k^(n-1)*binomial(n,k),k=1..n),n=1..22);

Rest[CoefficientList[Series[-2*LambertW[-x*E^(x/2)/2], {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jun 23 2016 *)

/* Egf A(x) = G(sqrt(G(x^2*exp(x)))) if G(sqrt(G(x^2*exp(-x)))) = x */
{a(n) = my(G=x); for(i=1,n, G = serreverse( sqrt( subst(G,x, x^2*exp(-x +O(x^n))) ) )); A = subst(G,x,sqrt(subst(G,x,x^2*exp(x +O(x^n))))); n!*polcoeff(A,n)}
for(n=1,30,print1(a(n),", ")) \\ Paul D. Hanna, Jun 06 2016

def A100526(n): return 2^(1-n)*sum( k^(n-1)*binomial(n, k) for k in range(1,n+1) )
[A100526(n) for n in range(1,40)] # G. C. Greubel, Mar 27 2023

a(n) = (1/2^(n-1))*Sum_{k=1..n} k^(n-1)*binomial(n, k).
a(n) = n*A007889(n-1).
E.g.f.: -2*LambertW(-x*exp(x/2)/2). - Paul D. Hanna, Jun 07 2016, after Vladeta Jovovic's formula in A038049
E.g.f.: G( sqrt( G(x^2*exp(x)) ) ), where G( sqrt( G(x^2*exp(-x)) ) ) = x, and G(x) is the e.g.f. of A273952. - Paul D. Hanna, Jun 06 2016
a(n) ~ sqrt(1 + LambertW(exp(-1)))*n^(n-1) / (2^(n-1) * exp(n) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Jun 23 2016

A038050 Number of labeled rooted trees with 3-colored leaves.

Original entry on oeis.org

3, 6, 45, 504, 7785, 153468, 3681909, 104126256, 3392064945, 125089571700, 5151335388309, 234322765501608, 11668410187187481, 631335472193760012, 36881146426978035765, 2313552152470193124192, 155107536736245864549345

Offset: 1

Christian G. Bower, Jan 04 1999

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 185 (3.1.83)

Index entries for sequences related to rooted trees
N. J. A. Sloane, Transforms

Cf. A000169, A029857.

Cf. A038049.

Rest[CoefficientList[Series[2*x-LambertW[-x*E^(2*x)], {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Oct 05 2013 *)

Divides by n and shifts left under exponential transform.
E.g.f.: 2*x - LambertW(-x*exp(2*x)). - Vladeta Jovovic, Mar 09 2003
a(n) = Sum_{k=0..n} (binomial(n, k)*2^k*(n-k)^(n-1)).
a(n) ~ sqrt(1+LambertW(2*exp(-1))) * (2*exp(-1)/LambertW(2*exp(-1)))^n * n^(n-1). - Vaclav Kotesovec, Oct 05 2013

A038053 Number of labeled planted trees with 2-colored leaves.

Original entry on oeis.org

0, 4, 12, 96, 1120, 17280, 330624, 7540736, 199544832, 6006988800, 202646118400, 7570772656128, 310240496517120, 13834761553313792, 666909048381112320, 34555424387503226880, 1915099718255940468736

Offset: 1

Christian G. Bower, Jan 04 1999

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for sequences related to rooted trees

CoefficientList[Series[(2+LambertW[-x*E^x])*(x-LambertW[-x*E^x])/(1+ LambertW[-x*E^x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Mar 29 2014 *)

x='x+O('x^30); concat([0], Vec(serlaplace( (2+lambertw(-x*exp(x))) *(x-lambertw(-x*exp(x)))/(1+lambertw(-x*exp(x))) ))) \\ G. C. Greubel, Sep 09 2018

A038049 shifted right and multiplied by n.
E.g.f. (for offset 0): (2+B(x))*(x-B(x))/(1+B(x)) where B(x) = LambertW(-x*exp(x)). - Vladeta Jovovic, Mar 08 2003
a(n) ~ sqrt(LambertW(exp(-1))+1) * n^(n-1) / (exp(n) * (LambertW(exp(-1)))^(n-1)). - Vaclav Kotesovec, Mar 29 2014

A360548 E.g.f. satisfies A(x) = x * exp( 2*(x + A(x)) ).

Original entry on oeis.org

0, 1, 8, 96, 1792, 46080, 1511424, 60325888, 2837970944, 153778913280, 9432255692800, 646039266656256, 48874810528235520, 4047655951598092288, 364221261622538141696, 35384754572803304325120, 3691411033400626898796544, 411569264258973944034361344

Offset: 0

Seiichi Manyama, Feb 11 2023

nonn

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

Cf. A100526, A216857, A360545.

Cf. A038049, A201595, A214225, A360547.

A360548 := proc(n)
    add((2*k)^(n-1)*binomial(n,k),k=1..n) ;
end proc:
seq(A360548(n),n=0..60) ; # R. J. Mathar, Mar 12 2023

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-2*x*exp(2*x))/2)))

a(n) = sum(k=1, n, (2*k)^(n-1)*binomial(n, k));

E.g.f.: A(x) = (-1/2) * LambertW(-2*x * exp(2*x)).
a(n) = Sum_{k=1..n} (2*k)^(n-1) * binomial(n,k) = 4^(n-1) * A100526(n).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 2^(n-1) * n^(n-1) / (LambertW(exp(-1))^n * exp(n)). - Vaclav Kotesovec, Feb 17 2023

cp's OEIS Frontend

A007889 Number of intransitive (or alternating, or Stanley) trees: vertices are [0,n] and for no i

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A216857 Number of connected functions from {1,2,...,n} into a subset of {1,2,...,n} that have a fixed point summed over all subsets.

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A349562 Number of labeled rooted forests with 2-colored leaves.

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A029856 Number of rooted trees with 2-colored leaves.

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A088789 E.g.f.: REVERT(2*x/(1+exp(x))) = Sum_{n>=0} a(n)*x^n/n!.

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A214225 E.g.f. satisfies: A(x) = x/(1 - tanh(A(x))).

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A088789 E.g.f.: REVERT(2x/(1+exp(x))) = Sum_{n>=0} a(n)x^n/n!.