A038037 -id:A038037 - cp's OEIS frontend

A138013 E.g.f. satisfies: A(x) = 1 - log(1 - x*A(x)).

1, 1, 3, 17, 146, 1694, 24834, 440586, 9180800, 219829536, 5948287560, 179508872520, 5978006444112, 217772950035120, 8614798644364080, 367768502385434640, 16852524904388586240, 825075552824125305600, 42981992589364756939008, 2373967488394457834095872

Offset: 0

Paul D. Hanna, Feb 27 2008

nonn

a(n) = A038037(n+1)/(n+1) for n>=0 where A038037(n) is the number of labeled rooted compound windmills (mobiles) with n nodes.

			E.g.f.: A(x) = 1 + x + 3x^2/2! + 17x^3/3! + 146x^4/4! + 1694x^5/5! + ...
where A(x) = 1 - log(1 - x*A(x)):
A(x) = 1 + x*A(x) + x^2*A(x)^2/2 + x^3*A(x)^3/3 +...+ x^n*A(x)^n/n +...

Cf. A038037.

CoefficientList[1 + InverseSeries[Series[(1-E^(-x))/(1+x), {x, 0, 20}], x],x] * Range[0, 20]! (* Vaclav Kotesovec, Dec 27 2013 *)

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

{a(n)=n!*polcoeff(1 + serreverse((1-exp(-x+x^2*O(x^n)))/(1+x +x*O(x^n))),n)}

E.g.f.: A(x) = (1/x)*Series_Reversion[ x/(1 - log(1-x)) ].
E.g.f.: A(x) = 1 + Series_Reversion( (1-exp(-x))/(1+x) ).
E.g.f. A(x) satisfies: exp(1 - A(x)) = 1 - x*A(x).
a(n) ~ sqrt(-1-LambertW(-1,-exp(-2))) * (-LambertW(-1,-exp(-2)))^n * n^(n-1) / exp(n). - Vaclav Kotesovec, Dec 27 2013
a(n) = sum(n!/(n+1-k)! * |stirling1(n,k)|, k=0..n). - Michael D. Weiner, Dec 23 2014

A029768 Number of increasing mobiles with n elements.

Original entry on oeis.org

0, 1, 1, 2, 7, 36, 245, 2076, 21059, 248836, 3356609, 50896380, 856958911, 15864014388, 320245960333, 7001257954796, 164792092647355, 4154906594518116, 111719929072986521, 3191216673497748444

Offset: 0

N. J. A. Sloane, Dec 11 1999

A labeled tree of size n is a rooted tree on n nodes that are labeled by distinct integers from the set {1,...,n}. An increasing tree is a labeled tree such that the sequence of labels along any branch starting at the root is increasing.
a(n) counts increasing trees with cyclically ordered branches.
a(n+1) counts the non-plane (where the subtrees stemming from a node are not ordered between themselves) increasing trees on n nodes where the nodes of outdegree k come in k+1 colors. An example is given below. The number of plane increasing trees on n nodes where the nodes of outdegree k come in k+1 colors is given by the triple factorial numbers A008544. - Peter Bala, Aug 30 2011
a(n+1)/a(n)/n tends to 1/A073003 = 1.676875... . - Vaclav Kotesovec, Mar 11 2014

			a(4) = 7: D^2[(1+x)*exp(x)] = exp(2*x)*(2*x^2+8*x+7). Evaluated at x = 0 this gives a(4) = 7. Denote the colors of the nodes by the letters a,b,c,.... The 7 possible trees on 3 nodes with nodes of outdegree k coming in k+1 colors are:
........................................................
...1a....1b....1a....1b........1a.......1b........1c....
...|.....|.....|.....|......../.\....../..\....../..\...
...2a....2b....2b....2a......2...3....2....3....2....3..
...|.....|.....|.....|..................................
...3.....3.....3.....3..................................
G.f. = x + x^2 + 2*x^3 + 7*x^4 + 36*x^5 + 245*x^6 + 2076*x^7 + 21059*x^8 + ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 392.

Vaclav Kotesovec, Table of n, a(n) for n = 0..400
C. G. Bower, Transforms
C. G. Bower, Transforms (2)
F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA], 2015.
Sergey Fomin and Grigory Mikhalkin, Labeled floor diagrams for plane curves, arXiv:0906.3828 [math.AG], 2009-2010.
Index entries for sequences related to mobiles

Cf. A032220, A038037, A055356, A008544, A145271.

S:= rhs(dsolve({diff(a(x),x) = log(1/(1-a(x)))+1,a(0)=0},a(x),series,order=101)):
seq(coeff(S,x,j)*j!,j=0..100); # Robert Israel, Apr 17 2015

Multinomial1[list_] := Apply[Plus, list]!/Apply[Times, (#1! & ) /@ list]; a[1]=1; a[n_]/;n>=2 := a[n] = Sum[Map[Multinomial1[ # ]Product[Map[a,# ]]/Length[ # ]&,Compositions[n-1]]]; Table[a[n],{n,8}] (* David Callan, Nov 29 2007 *)
nmax=20; b = ConstantArray[0,nmax]; b[[1]]=0; b[[2]]=1; Do[b[[n+1]] = b[[n]] + Sum[Binomial[n-2,i]*b[[i+1]]*b[[n-i+1]],{i,1,n-2}],{n,2,nmax-1}]; b (* Vaclav Kotesovec after Vladimir Kruchinin, Mar 11 2014 *)
terms = 20; A[x_] := x; Do[A[x_] = Integrate[(1 + A[x])*Exp[A[x] + O[x]^j], x] + O[x]^j // Normal // Simplify, {j, 1, terms - 1}]; Join[{0, 1}, CoefficientList[A[x], x]*Range[0, terms - 2]! // Rest] (* Jean-François Alcover, May 22 2014, updated Jan 12 2018 (after PARI script by Michael Somos) *)

{a(n) = my(A = x + O(x^2)); if( n<2, n==1, n--; for(k=1, n-1, A = intformal( (1 + A) * exp(A)));  n! * polcoeff(A, n))}; /* Michael Somos, Apr 17 2015 */
for(n=1,30,print1(a(n),", "))

seq(N) = {
  my(a = vector(N)); a[1] = 1;
  for (n = 2, N, a[n] = a[n-1] + sum(k=1, n-2, binomial(n-2, k)*a[k]*a[n-k]));
  concat(0, a);
};
seq(19)
\\ test: N=200; y=serconvol(Ser(seq(N),'x), exp('x+O('x^N))); y' == y''*(1-y)
\\ Gheorghe Coserea, Jun 26 2018

Bergeron et al. give several formulas. Shifts left under "CIJ" (necklace, indistinct, labeled) transform.
E.g.f.: A(x) =
  x + (1/2)*x^2 + (1/3)*x^3 + (7/24)*x^4 + (3/10)*x^5 + (49/144)*x^6 + (173/420)*x^7 + (21059/40320)*x^8 + (8887/12960)*x^9 + ...
and satisfies the differential equation A'(x)=log(1/(1-A(x)))+1. - Vladimir Kruchinin, Jan 22 2011
E.g.f. A(x) satisfies: A''(x) = A'(x) * exp(A'(x)-1). - Paul D. Hanna, Apr 17 2015
From Robert Israel, Apr 17 2015 (Start):
E.g.f. A(x) satisfies e*(Ei(1,A'(x)) - Ei(1,1)) = integral(s = 1 .. A'(x), exp(1-s)/s ds) = -x.
a(n) = e^(1-n)*limit(w -> 1, (d^(n-2)/dw^(n-2))(((w-1)/(Ei(1,1)-Ei(1,w)))^(n-1))) for n >= 2. (End)
a(n) = sum(i=1..n-2,binomial(n-2,i)*a(i)*a(n-i))+a(n-1), a(0)=0, a(1)=1. - Vladimir Kruchinin, Jan 24 2011
The following remarks refer to the interpretation of this sequence as counting increasing trees where the nodes of outdegree k come in k+1 colors. Thus we work with the generating function B(x) = A'(x)-1 = x + 2*x^2/2!+7*x^3/3!+36*x^4/4!+.... The degree function phi(x) (see [Bergeron et al.] for definition) for this variety of trees is phi(x) = 1+2*x+3*x^2/2!+4*x^3/3!+5*x^4/4!+... = (1+x)*exp(x). The generating function B(x) satisfies the autonomous differential equation B' = phi(B(x)) with initial condition B(0) = 0. It follows that the inverse function B(x)^(-1) may be expressed as an integral B(x)^(-1) = int {t = 0..x} 1/phi(t) dt = int {t = 0..x} exp(-t)/(1+t) dt. Applying [Dominici, Theorem 4.1] to invert the integral produces the result B(x) = sum {n>=1} D^(n-1)[(1+x)*exp(x)](0)*x^n/n!, where the nested derivative D^n[f](x) of a function f(x) is defined recursively as D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0. Thus a(n+1) = D^(n-1)[(1+x)*exp(x)](0). - Peter Bala, Aug 30 2011

More terms from Christian G. Bower

A032200 Number of rooted compound windmills (mobiles) of n nodes.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 51, 128, 345, 940, 2632, 7450, 21434, 62174, 182146, 537369, 1596133, 4767379, 14312919, 43162856, 130695821, 397184252, 1211057426, 3703794849, 11358759346, 34923477315, 107627138308, 332404636811

Offset: 1

Christian G. Bower

Also the number of locally necklace plane trees with n nodes, where a plane tree is locally necklace if the sequence of branches directly under any given node is lexicographically minimal among its cyclic permutations. - Gus Wiseman, Sep 05 2018

			From _Gus Wiseman_, Sep 05 2018: (Start)
The a(5) = 9 locally necklace plane trees:
  ((((o))))
  (((oo)))
  ((o(o)))
  (o((o)))
  ((o)(o))
  ((ooo))
  (o(oo))
  (oo(o))
  (oooo)
(End)

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 241 (3.3.84).

Andrew Howroyd, Table of n, a(n) for n = 1..200
C. G. Bower, Transforms (2)
Index entries for sequences related to mobiles

Cf. A029768, A038037, A055340.

Cf. A000108, A007853, A032171, A254040, A304173, A304175, A317852.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
neckplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[neckplane/@c],neckQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[neckplane[n]],{n,10}] (* Gus Wiseman, Sep 05 2018 *)

CIK(p,n)={sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=O(1));for(i=1, n, p=1+CIK(x*p, i)); Vec(p)} \\ Andrew Howroyd, Jun 20 2018

Shifts left under "CIK" (necklace, indistinct, unlabeled) transform.

A055349 Triangle of labeled mobiles (circular rooted trees) with n nodes and k leaves.

Original entry on oeis.org

1, 2, 0, 6, 3, 0, 24, 36, 8, 0, 120, 360, 220, 30, 0, 720, 3600, 4200, 1500, 144, 0, 5040, 37800, 71400, 47250, 11508, 840, 0, 40320, 423360, 1176000, 1234800, 545664, 98784, 5760, 0, 362880, 5080320, 19474560, 29635200, 20469456, 6618528, 940896, 45360, 0

Offset: 1

Christian G. Bower, May 15 2000

			Triangle begins:
     1;
     2,     0;
     6,     3,     0;
    24,    36,     8,     0;
   120,   360,   220,    30,     0;
   720,  3600,  4200,  1500,   144,   0;
  5040, 37800, 71400, 47250, 11508, 840, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
Index entries for sequences related to mobiles

Row sums give A038037.
Columns 1..8 are A000142, A055303, A055350, A055351, A055352, A055353, A055354, A055355.
Cf. A055340, A055356, A055363.

T[rows_] := {{1}}~Join~((cc = CoefficientList[#, y]; Append[Rest[cc], 0] * Length[cc]!)& /@ (CoefficientList[InverseSeries[x/(y-Log[1-x + O[x]^rows] ), x], x][[3;;]]));
T[9] // Flatten (* Jean-François Alcover, Oct 31 2019 *)

A(n)={my(v=Vec(serlaplace(serreverse(x/(y - log(1-x + O(x^n))))))); vector(#v, i, Vecrev(v[i]/y, i))}
{ my(T=A(10)); for(i=1, #T, print(T[i])) } \\ Andrew Howroyd, Sep 23 2018

E.g.f. satisfies A(x, y) = x*y - x*log(1-A(x, y)). [Corrected by Sean A. Irvine, Mar 19 2022]

A177380 E.g.f. satisfies: A(x) = 1+x + x*log(A(x)).

Original entry on oeis.org

1, 1, 2, 3, -4, -50, -36, 2058, 10800, -131616, -1975680, 7741800, 417480480, 1307617584, -101626746144, -1284067345680, 25419094122240, 791333924647680, -3900043588999680, -472446912421801728, -3183064994777932800

Offset: 0

Paul D. Hanna, May 14 2010

sign

The signs have a complex structure; are they periodic after some point?

			E.g.f: A(x) = 1 + x + 2*x^2/2! + 3*x^3/3! - 4*x^4/4! - 50*x^5/5! +...
log(A(x)) = 2*x/2! + 3*x^2/3! - 4*x^3/4! - 50*x^4/5! - 36*x^5/5! +...
...
Coefficients in the initial powers of A(x) begin:
[1,(1),(1), 1/2, -1/6, -5/12, -1/20, 49/120, 15/56, -457/1260,...];
[1, 2,(3),(3), 5/3, -1/6, -61/60, -17/60, 272/315, 451/630,...];
[1, 3, 6,(17/2),(17/2), 21/4, 3/5, -83/40, -187/168, 115/84,...];
[1, 4, 10, 18,(73/3),(73/3), 163/10, 131/30, -261/70, -1093/315,...];
[1, 5, 15, 65/2, 325/6,(847/12),(847/12), 1205/24, 9551/504,...];
[1, 6, 21, 53, 104, 327/2,(4139/20),(4139/20), 6469/42, 7414/105,...];
[1, 7, 28, 161/2, 1085/6, 3955/12, 4949/10,(24477/40),(24477/40),...];
[1, 8, 36, 116, 878/3, 1810/3, 15569/15, 7509/5,(114760/63),(114760/63), ...]; ...
where the coefficients in parenthesis illustrate the property
that the coefficients of x^n and x^(n+1) in A(x)^n are equal:
[x^n] A(x)^n = [x^(n+1)] A(x)^n = A138013(n)/(n-1)!,
where G(x) = e.g.f. of A138013 begins:
G(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 146*x^4/4! + 1694*x^5/5! + ...
and satisfies: exp(1 - G(x)) = 1 - x*G(x).

Vaclav Kotesovec, Table of n, a(n) for n = 0..420

Cf. A177379, A138013, A038037, A238274.

CoefficientList[1+InverseSeries[Series[x/(1 + Log[1+x]), {x, 0, 20}], x],x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 11 2014 *)

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

{a(n)=local(A=1+x);for(i=1,n,A=1+x+x*log(A+O(x^n)));n!*polcoeff(A,n)}

E.g.f.: A(x) = 1 + Series_Reversion( x/(1 + log(1+x)) ).
...
Let G(x) = e.g.f. of A138013, then G(x) and A(x) satisfy:
(1) [x^n] A(x)^n = [x^(n+1)] A(x)^n = A138013(n)/(n-1)! for n>=1;
(2) A(x/(1 - x*G(x))) = 1/(1 - x*G(x));
(3) G(x) = 1 - log(1 - x*G(x)) = Series_Reversion(x/(1-log(1-x)))/x.
...
Let F(x) = e.g.f. of A177379, then F(x) and A(x) satisfy:
(4) [x^n] A(x)^(n+1)/(n+1) = A177379(n)/n! for n>=0;
(5) A(x*F(x)) = F(x) and F(x/A(x)) = A(x);
(6) F(x) = 1/(1 - x*G(x)) = 1/(1 - Series_Reversion(x/(1-log(1-x)))).
Lim sup n->infinity (|a(n)|/n!)^(1/n) = abs(LambertW(-1)) = 1.3745570107437... (see A238274). - Vaclav Kotesovec, Jan 11 2014

A108527 Number of labeled mobiles (cycle rooted trees) with n generators.

Original entry on oeis.org

1, 3, 20, 229, 3764, 80383, 2107412, 65436033, 2347211812, 95492023811, 4344109422388, 218499395486909, 12039757564700644, 721239945304498215, 46669064731537444820, 3243864647191662324601, 241046155271316751794596

Offset: 1

Christian G. Bower, Jun 07 2005

nonn

A generator is a leaf or a node with just one child.

Vaclav Kotesovec, Table of n, a(n) for n = 1..250
Index entries for sequences related to mobiles

Cf. A108521-A108529, A038037, A032188.

nmax=20; c[0]=0; A[x_]:=Sum[c[k]*x^k/k!,{k,0,nmax}]; Array[c,nmax]/.Solve[Rest[CoefficientList[Series[x-1-Log[1-A[x]]-(2-x)*A[x],{x,0,nmax}],x]]==0][[1]] (* Vaclav Kotesovec, Mar 28 2014 *)

{a(n)=local(A=x+O(x^n)); for(i=0, n, A=intformal((1-A^2)/(1-x-2*A+x*A)+O(x^n))); n!*polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 28 2014

E.g.f. satisfies: (2-x)*A(x) = x - 1 - log(1-A(x)).

a(n) ~ c * n^(n-1) / (exp(n) * r^n), where r = 0.20846306198165450115960050053484328028... and c = 0.3060161306524907981116283162103879... - Vaclav Kotesovec, Mar 28 2014

A242769 Decimal expansion of the positive solution to the equation x/(1-x) = 1+log(1/(1-x)), an auxiliary constant associated with the problem of enumeration of trees by inversions.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 22 2014

			0.6821555671006273161671552623790508330038681...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.6, p. 303.

I. M. Gessel, B. E. Sagan, Y. N. Yeh, Enumeration of Trees By Inversions, 1999, p. 20.

Cf. A038037.

mu = x /. FindRoot[x/(1-x) == 1+Log[1/(1-x)], {x, 1/2}, WorkingPrecision -> 105]; RealDigits[mu, 10, 100] // First

A174632 Partial sums of A029768.

Original entry on oeis.org

0, 1, 2, 4, 11, 47, 292, 2368, 23427, 272263, 3628872, 54525252, 911484163, 16775498551, 337021458884, 7338279413680, 172130372061035, 4327036966579151, 116046966039565672, 3307263639537314116

Offset: 0

Jonathan Vos Post, Mar 24 2010

nonn

Partial sums of number of increasing mobiles with n elements. In an increasing rooted tree, nodes are numbered and numbers increase as you move away from root. The subsequence of primes in this partial sum begins: 2, 11, 47, 272263.

			a(x) = 0 + 1 + 1 + 2 + 7 + 36 + 245 + 2076 + 21059 + 248836 = 272263 is prime.

Robert Israel, Table of n, a(n) for n = 0..416

Cf. A029768, A032220, A038037, A055356.

S:= rhs(dsolve({diff(a(x), x) = log(1/(1-a(x)))+1, a(0)=0}, a(x), series, order=31)):
L:= [seq(coeff(S, x, j)*j!, j=0..30)]:
ListTools:-PartialSums(L); # Robert Israel, Dec 21 2017

a(n) = SUM[i=o..n] A029768(i).

A243395 Decimal expansion of 'xiHat', a constant used in the asymptotic evaluation of e.g.f. coefficients for the number of labeled mobiles.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 04 2014

			1.15741980381928026448263545084...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.6 Otter's tree enumeration constants, p. 303.

Cf. A038037, A242769, A243396.

digits = 104; mu = x /. FindRoot[x/(1 - x) == 1 + Log[1/(1 - x)], {x, 1/2}, WorkingPrecision -> digits + 5] ; xiHat = E^-1*(1 - mu)^-1; RealDigits[xiHat, 10, digits] // First

A038037 = Numerator of n-th coefficient ~ etaHat * xiHat^n * n^(n-1), where etaHat is A243396.

A243396 Decimal expansion of 'etaHat', a constant used in the asymptotic evaluation of e.g.f. coefficients for the number of labeled mobiles.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 04 2014

			0.46563864677907573710997591026429...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.6 Otter's tree enumeration constants, p. 303.

Cf. A038037, A242769, A243395.

digits = 104; mu = x /. FindRoot[x/(1 - x) == 1 + Log[1/(1 - x)], {x, 1/2}, WorkingPrecision -> digits + 5] ; etaHat = Sqrt[mu*(1 - mu)]; RealDigits[etaHat, 10, digits] // First

A038037 = Numerator of n-th coefficient ~ etaHat * xiHat^n * n^(n-1), where xiHat is A243395.

cp's OEIS Frontend

A138013 E.g.f. satisfies: A(x) = 1 - log(1 - x*A(x)).

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A029768 Number of increasing mobiles with n elements.

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A032200 Number of rooted compound windmills (mobiles) of n nodes.

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A055349 Triangle of labeled mobiles (circular rooted trees) with n nodes and k leaves.

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A177380 E.g.f. satisfies: A(x) = 1+x + x*log(A(x)).

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A108527 Number of labeled mobiles (cycle rooted trees) with n generators.

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A242769 Decimal expansion of the positive solution to the equation x/(1-x) = 1+log(1/(1-x)), an auxiliary constant associated with the problem of enumeration of trees by inversions.

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