A000669 -id:A000669 - cp's OEIS frontend

A060356 Expansion of e.g.f.: -LambertW(-x/(1+x)).

0, 1, 0, 3, 4, 65, 306, 4207, 38424, 573057, 7753510, 134046671, 2353898196, 47602871329, 1013794852266, 23751106404495, 590663769125296, 15806094859299329, 448284980183376078, 13515502344669830287

Offset: 0

Vladeta Jovovic, Apr 01 2001

Also the number of labeled lone-child-avoiding rooted trees with n nodes. A rooted tree is lone-child-avoiding if it has no unary branchings, meaning every non-leaf node covers at least two other nodes. The unlabeled version is A001678(n + 1). - Gus Wiseman, Jan 20 2020

			From _Gus Wiseman_, Dec 31 2019: (Start)
Non-isomorphic representatives of the a(7) = 4207 trees, written as root[branches], are:
  1[2,3[4,5[6,7]]]
  1[2,3[4,5,6,7]]
  1[2[3,4],5[6,7]]
  1[2,3,4[5,6,7]]
  1[2,3,4,5[6,7]]
  1[2,3,4,5,6,7]
(End)

Harry J. Smith, Table of n, a(n) for n = 0..100
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Cf. A052871, A060313.
Cf. A008297.
Column k=0 of A231602.
The unlabeled version is A001678(n + 1).
The case where the root is fixed is A108919.
Unlabeled rooted trees are counted by A000081.
Lone-child-avoiding rooted trees with labeled leaves are A000311.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Singleton-reduced rooted trees are counted by A330951.
Cf. A000669, A004111, A005121, A048816, A292504, A316651, A316652, A318231, A318813, A330465, A330624.

List([0..20],n->Sum([1..n],k->(-1)^(n-k)*Factorial(n)/Factorial(k) *Binomial(n-1,k-1)*k^(k-1))); # Muniru A Asiru, Feb 19 2018

seq(coeff(series( -LambertW(-x/(1+x)), x, n+1), x, n)*n!, n = 0..20); # G. C. Greubel, Mar 16 2020

CoefficientList[Series[-LambertW[-x/(1+x)], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
a[n_]:=If[n==1,1,n*Sum[Times@@a/@Length/@stn,{stn,Select[sps[Range[n-1]],Length[#]>1&]}]];
Array[a,10] (* Gus Wiseman, Dec 31 2019 *)

{ for (n=0, 100, f=n!; a=sum(k=1, n, (-1)^(n - k)*f/k!*binomial(n - 1, k - 1)*k^(k - 1)); write("b060356.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 04 2009

my(x='x+O('x^20)); concat([0], Vec(serlaplace(-lambertw(-x/(1+x))))) \\ G. C. Greubel, Feb 19 2018

a(n) = Sum_{k=1..n} (-1)^(n-k)*n!/k!*binomial(n-1, k-1)*k^(k-1). a(n) = Sum_{k=0..n} Stirling1(n, k)*A058863(k). - Vladeta Jovovic, Sep 17 2003
a(n) ~ n^(n-1) * (1-exp(-1))^(n+1/2). - Vaclav Kotesovec, Nov 27 2012
a(n) = n * A108919(n). - Gus Wiseman, Dec 31 2019

A316651 Number of series-reduced rooted trees with n leaves spanning an initial interval of positive integers.

Original entry on oeis.org

1, 2, 12, 112, 1444, 24086, 492284, 11910790, 332827136, 10546558146, 373661603588, 14636326974270, 628032444609396, 29296137817622902, 1476092246351259964, 79889766016415899270, 4622371378514020301740, 284719443038735430679268, 18601385258191195218790756

Offset: 1

Gus Wiseman, Jul 09 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches.

			The a(3) = 12 trees:
  (1(11)), (111),
  (1(12)), (2(11)), (112),
  (1(22)), (2(12)), (122),
  (1(23)), (2(13)), (3(12)), (123).

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000081, A000311, A000669, A001678, A005804, A034691, A141268, A292504.
Cf. A316652, A316653, A316654, A316655, A316656, A319369.
Row sums of A319376.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(A(i, k)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
A:= (n, k)-> `if`(n<2, n*k, b(n, n-1, k)):
a:= n-> add(add(A(n, k-j)*(-1)^j*binomial(k, j), j=0..k-1), k=1..n):
seq(a(n), n=1..20);  # Alois P. Heinz, Sep 18 2018

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
gro[m_]:=If[Length[m]==1,m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Sum[Length[gro[m]],{m,allnorm[n]}],{n,5}]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0,
     Sum[Binomial[A[i, k] + j - 1, j] b[n - i*j, i - 1, k], {j, 0, n/i}]]];
A[n_, k_] := If[n < 2, n*k, b[n, n - 1, k]];
a[n_] := Sum[Sum[A[n, k-j]*(-1)^j*Binomial[k, j], {j, 0, k-1}], {k, 1, n}];
Array[a, 20] (* Jean-François Alcover, May 09 2021, after Alois P. Heinz *)

\\ here R(n,k) is A000669, A050381, A220823, ...
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n,k)={my(v=[k]); for(n=2, n, v=concat(v, EulerT(concat(v,[0]))[n])); v}
seq(n)={sum(k=1, n, R(n,k)*sum(r=k, n, binomial(r,k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Sep 14 2018

From Vaclav Kotesovec, Sep 18 2019: (Start)
a(n) ~ c * d^n * n^(n-1), where d = 1.37392076830840090205551979... and c = 0.41435722857311602982846...
a(n) ~ 2*log(2)*A326396(n)/n. (End)

Terms a(9) and beyond from Andrew Howroyd, Sep 14 2018

A006024 Number of labeled mating graphs with n nodes. Also called point-determining graphs.

Original entry on oeis.org

1, 1, 1, 4, 32, 588, 21476, 1551368, 218608712, 60071657408, 32307552561088, 34179798520396032, 71474651351939175424, 296572048493274368856832, 2448649084251501449508762880, 40306353989748719650902623919616

Offset: 0

Simon Plouffe

nonn

A mating graph is one in which no two vertices have identical adjacencies with the other vertices. - Ronald C. Read and Vladeta Jovovic, Feb 10 2003

Also number of (n-1)-node labeled mating graphs allowing loops and without isolated nodes. - Vladeta Jovovic, Mar 08 2008

			Consider the square (cycle of length 4) on vertices 1, 2, 3 and 4 in that order. Join a fifth vertex (5) to vertices 1, 3 and 4. The resulting graph is not a mating graph since vertices 1 and 3 both have the set {2, 4, 5} as neighbors. If we delete the edge (1,5) then the resulting graph is a mating graph: the neighborhood sets for vertices 1, 2, 3, 4 and 5 are respectively {2,4}, {1,3}, {2,4,5}, {1,3,5} and {3,4} - all different.

R. C. Read, The Enumeration of Mating-Type Graphs. Report CORR 89-38, Dept. Combinatorics and Optimization, Univ. Waterloo, 1989.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..50
I. M. Gessel and J. Li, Enumeration of Point-Determining Graphs, arXiv:0705.0042 [math.CO], 2007-2009.
I. M. Gessel and J. Li, Enumeration of point-determining graphs, J. Combinatorial Theory Ser. A 118 (2011), 591-612.
R. C. Read, The Enumeration of Mating-Type Graphs, Dept. Combinatorics and Optimization, Univ. Waterloo, Oct 1989. (Annotated scanned copy)
D. Sumner, Point determination in graphs, Discrete Mathematics 5 (1973), 179-187.

Cf. A006025.
Cf. bi-point-determining graphs: labeled A129583, unlabeled A129584; connected bi-point-determining graphs: labeled A129585, unlabeled A129586; phylogenetic trees: labeled A000311, unlabeled A000669.
Cf. A007833, A079306 (connected)

a[n_] := Sum[StirlingS1[n, k] 2^Binomial[k, 2], {k, 0, n}];
Array[a, 15] (* Jean-François Alcover, Jul 25 2018 *)

a(n)=n!*polcoeff(sum(k=0,n,2^(k*(k-1)/2)*log(1+x+x*O(x^n))^k/k!),n) \\ Paul D. Hanna, May 20 2009

a(n) = Sum_{k=0..n} Stirling1(n, k)*2^binomial(k, 2). - Ronald C. Read and Vladeta Jovovic, Feb 10 2003

E.g.f.: Sum_{n>=0} 2^(n(n-1)/2)*log(1+x)^n/n!. - Paul D. Hanna, May 20 2009

More terms from Ronald C. Read and Vladeta Jovovic, Feb 10 2003

a(0)=1 prepended by Andrew Howroyd, Sep 09 2018

A006351 Number of series-parallel networks with n labeled edges. Also called yoke-chains by Cayley and MacMahon.

Original entry on oeis.org

1, 2, 8, 52, 472, 5504, 78416, 1320064, 25637824, 564275648, 13879795712, 377332365568, 11234698041088, 363581406419456, 12707452084972544, 477027941930515456, 19142041172838025216, 817675811320888020992, 37044610820729973813248, 1774189422608238694776832

Offset: 1

N. J. A. Sloane

For a simple relationship to series-reduced rooted trees, partitions of n, and phylogenetic trees among other combinatoric constructs, see comments in A000311. - Tom Copeland, Jan 06 2021

			D^3(1) = (12*x^2+56*x+52)/(x-1)^6. Evaluated at x = 0 this gives a(4) = 52.
a(3) = 8: The 8 possible increasing plane trees on 3 vertices with vertices of outdegree k >= 1 coming in 2 colors, B or W, are
.......................................................
.1B..1B..1W..1W.....1B.......1W........1B........1W....
.|...|...|...|...../.\....../..\....../..\....../..\...
.2B..2W..2B..2W...2...3....2....3....3....2....3....2..
.|...|...|...|.........................................
.3...3...3...3.........................................
G.f. = x + 2*x^2 + 8*x^3 + 52*x^4 + 472*x^5 + 5504*x^6 + 78416*x^7 + ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 417.
P. A. MacMahon, Yoke-trains and multipartite compositions in connexion with the analytical forms called "trees", Proc. London Math. Soc. 22 (1891), 330-346; reprinted in Coll. Papers I, pp. 600-616. Page 333 gives A000084 = 2*A000669.
P. A. MacMahon, The combination of resistances, The Electrician, 28 (1892), 601-602; reprinted in Coll. Papers I, pp. 617-619.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 142.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.40(a), S(x).

Alois P. Heinz, Table of n, a(n) for n = 1..200
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.
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.
Peter J. Cameron, Counting two-graphs related to trees, Elec. J. Combin., Vol. 2, #R4. [From _Aaron Meyerowitz_, Sep 30 2010]
Frédéric Chapoton, Florent Hivert, and Jean-Christophe Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013
Diego Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052 [math.CA], 2005.
Brian Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.5.1 ), Ph. D. dissertation, Brandeis Univ., 2008.
Steven R. Finch, Series-parallel networks
Steven R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
ISCGI, Cograph graphs
Song He, Fei Teng, and Yong Zhang, String Correlators: Recursive Expansion, Integration-by-Parts and Scattering Equations, arXiv:1907.06041 [hep-th], 2019. Also in Journal of High Energy Physics (2019), 2019:85.
Walter Knödel, Über Zerfällungen, Monatsh. Math., 55 (1951), 20-27.
Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.
Z. A. Lomnicki, Two-terminal series-parallel networks, Adv. Appl. Prob., 4 (1972), 109-150.
Michael D. Moffitt, Search Strategies for Topological Network Optimization, Proceedings of the AAAI Conference on Artificial Intelligence, 36(9) (2022), 10299-10308.
John W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the e.g.f. U(x)).
N. J. A. Sloane, Transforms
Index entries for reversions of series
Index entries for sequences mentioned in Moon (1987)
Index entries for sequences related to trees [From _Aaron Meyerowitz_, Sep 30 2010]

Cf. A000311, A000084 (for unlabeled case), A032188. A140945.

Cf. A133314, A134991, A135494.

read transforms; t1 := 2*ln(1+x)-x; t2 := series(t1,x,10); t3 := seriestoseries(t2,'revogf'); t4 := SERIESTOLISTMULT(%);
# N denotes all series-parallel networks, S = series networks, P = parallel networks;
spec := [ N, N=Union(Z,S,P),S=Set(Union(Z,P),card>=2), P=Set(Union(Z,S), card>=2)}, labeled ]: A006351 := n->combstruct[count](spec,size=n);
A006351 := n -> add(combinat[eulerian2](n-1,k)*2^(n-k-1),k=0..n-1):
seq(A006351(n), n=1..18); # Peter Luschny, Nov 16 2012

max = 18; f[x_] := 2*Log[1+x]-x; Rest[ CoefficientList[ InverseSeries[ Series[ f[x], {x, 0, max}], x], x]]*Range[max]! (* Jean-François Alcover, Nov 25 2011 *)

a(n):=if n=1 then 1 else ((n-1)!*sum(binomial(n+k-1,n-1)* sum((-1)^(j)*binomial(k,j)*sum((binomial(j,l)*(j-l)!*2^(j-l)*(-1)^l* stirling1(n-l+j-1,j-l))/(n-l+j-1)!,l,0,j),j,1,k),k,1,n-1)); /* Vladimir Kruchinin, Jan 24 2012 */

x='x+O('x^66); Vec(serlaplace(serreverse( 2*log(1+x) - 1*x ))) \\ Joerg Arndt, May 01 2013

# uses[eulerian2 from A201637]
def A006351(n): return add(A201637(n-1, k)*2^(n-k-1) for k in (0..n-1))
[A006351(n) for n in (1..18)]  # Peter Luschny, Nov 16 2012

For n >= 2, A006351(n) = 2*A000311(n) = A005640(n)/2^n. Row sums of A140945.
E.g.f. is reversion of 2*log(1+x)-x.
Also exponential transform of A000311, define b by 1+sum b_n x^n / n! = exp ( 1 + sum a_n x^n /n!).
E.g.f.: A(x), B(x)=x*A(x) satisfies the differential equation B'(x)=(1+B(x))/(1-B(x)). - Vladimir Kruchinin, Jan 18 2011
From Peter Bala, Sep 05 2011: (Start)
The generating function A(x) satisfies the autonomous differential equation A'(x) = (1+A)/(1-A) with A(0) = 0. Hence the inverse function A^-1(x) = int {t = 0..x} (1-t)/(1+t) = 2*log(1+x)-x, which yields A(x) = -1-2*W(-1/2*exp((x-1)/2)), where W is the Lambert W function.
The expansion of A(x) can be found by inverting the above integral using the method of [Dominici, Theorem 4.1] to arrive at the result a(n) = D^(n-1)(1) evaluated at x = 0, where D denotes the operator g(x) -> d/dx((1+x)/(1-x)*g(x)). Compare with A032188.
Applying [Bergeron et al., Theorem 1] to the result x = int {t = 0..A(x)} 1/phi(t), where phi(t) = (1+t)/(1-t) = 1 + 2*t + 2*t^2 + 2*t^3 + ..., leads to the following combinatorial interpretation for the sequence: a(n) gives the number of plane increasing trees on n vertices where each vertex of outdegree k >=1 can be in one of 2 colors. An example is given below. (End)
A134991 gives (b.+c.)^n = 0^n , for (b_n)=A000311(n+1) and (c_0)=1, (c_1)=-1, and (c_n)=-2* A000311(n) = -A006351(n) otherwise. E.g., umbrally, (b.+c.)^2 = b_2*c_0 + 2 b_1*c_1 + b_0*c_2 =0. - Tom Copeland, Oct 19 2011
G.f.: 1/S(0) where S(k) = 1 - x*(k+1) - x*(k+1)/S(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 18 2011
a(n) = ((n-1)!*sum(k=1..n-1, C(n+k-1,n-1)*sum(j=1..k, (-1)^(j)*C(k,j)* sum(l=0..j, (C(j,l)*(j-l)!*2^(j-l)*(-1)^l*stirling1(n-l+j-1,j-l))/ (n-l+j-1)!)))), n>1, a(1)=1. - Vladimir Kruchinin, Jan 24 2012
E.g.f.: A(x) = exp(B(x))-1 where B(x) is the e.g.f. of A000311. - Vladimir Kruchinin, Sep 25 2012
a(n) = sum_{k=0..n-1} A201637(n-1,k)*2^(n-k-1). - Peter Luschny, Nov 16 2012
G.f.: -1 + 2/Q(0), where Q(k)= 1 - k*x - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
a(n) ~ sqrt(2)*n^(n-1)/((2*log(2)-1)^(n-1/2)*exp(n)). - Vaclav Kotesovec, Jul 17 2013
G.f.: Q(0)/(1-x), where Q(k) = 1 - x*(k+1)/( x*(k+1) - (1 -x*(k+1))*(1 -x*(k+2))/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 10 2013
a(1) = 1; a(n) = a(n-1) + Sum_{k=1..n-1} binomial(n-1,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Aug 28 2020
Conjecture: a(n) = A379459(n-2,0) = A379460(n-1,0) for n > 1 with a(1) = 1. - Mikhail Kurkov, Jan 16 2025

A316652 Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

Original entry on oeis.org

1, 2, 9, 69, 623, 7793, 110430, 1906317, 36833614, 816101825, 19925210834, 541363267613, 15997458049946, 515769374925576, 17905023985615254, 669030297769291562, 26689471638523499483, 1134895275721374771655, 51161002326406795249910, 2440166138715867838359915

Offset: 1

Gus Wiseman, Jul 09 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches.

			The a(3) = 9 trees:
(1(11)), (111),
(1(12)), (2(11)), (112),
(1(23)), (2(13)), (3(12)), (123).

Cf. A000081, A000311, A000669, A001678, A005804, A141268, A181821, A292504, A304660.

Cf. A316651, A316653, A316654, A316655, A316656.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
gro[m_]:=If[Length[m]==1,m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
Table[Sum[Length[gro[m]],{m,Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]}],{n,4}]

\\ See A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sExp(x*Ser(v[1..n])), n )); x*Ser(v)}
StronglyNormalLabelingsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Jan 04 2021

Terms a(10) and beyond from Andrew Howroyd, Jan 04 2021

A320160 Number of series-reduced balanced rooted trees whose leaves form an integer partition of n.

Original entry on oeis.org

1, 2, 3, 6, 9, 19, 31, 63, 110, 215, 391, 773, 1451, 2879, 5594, 11173, 22041, 44136, 87631, 175155, 348186, 694013, 1378911, 2743955, 5452833, 10853541, 21610732, 43122952, 86192274, 172753293, 347114772, 699602332, 1414033078, 2866580670, 5826842877, 11874508385

Offset: 1

Gus Wiseman, Oct 06 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root.

Also the number of balanced unlabeled phylogenetic rooted trees with n leaves.

			The a(1) = 1 through a(6) = 19 rooted trees:
  1  2     3      4           5            6
     (11)  (12)   (13)        (14)         (15)
           (111)  (22)        (23)         (24)
                  (112)       (113)        (33)
                  (1111)      (122)        (114)
                  ((11)(11))  (1112)       (123)
                              (11111)      (222)
                              ((11)(12))   (1113)
                              ((11)(111))  (1122)
                                           (11112)
                                           (111111)
                                           ((11)(13))
                                           ((11)(22))
                                           ((12)(12))
                                           ((11)(112))
                                           ((12)(111))
                                           ((11)(1111))
                                           ((111)(111))
                                           ((11)(11)(11))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000081, A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A319312.

Cf. A316624, A320154, A320155, A320169, A320173, A320179.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
phy2[labs_]:=If[Length[labs]==1,labs,Union@@Table[Sort/@Tuples[phy2/@ptn],{ptn,Select[mps[Sort[labs]],Length[#1]>1&]}]];
Table[Sum[Length[Select[phy2[ptn],SameQ@@Length/@Position[#,_Integer]&]],{ptn,IntegerPartitions[n]}],{n,8}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(u=vector(n, n, 1), v=vector(n)); while(u, v+=u; u=EulerT(u)-u); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(14) and beyond from Andrew Howroyd, Oct 25 2018

A050381 Number of series-reduced planted trees with n leaves of 2 colors.

Original entry on oeis.org

2, 3, 10, 40, 170, 785, 3770, 18805, 96180, 502381, 2667034, 14351775, 78096654, 429025553, 2376075922, 13252492311, 74372374366, 419651663108, 2379399524742, 13549601275893, 77460249369658, 444389519874841

Offset: 1

Christian G. Bower, Nov 15 1999

nonn

Consider the free algebraic system with two commutative associative operators (x+y) and (x*y) and two generators A,B. The number of elements with n occurrences of the generators is 2*a(n) if n>1, and the number of generators if n=1. - Michael Somos, Aug 07 2017
From Gus Wiseman, Feb 07 2020: (Start)
Also the number of semi-lone-child-avoiding rooted trees with n leaves. Semi-lone-child-avoiding means there are no vertices with exactly one child unless that child is an endpoint/leaf. For example, the a(1) = 2 through a(3) = 10 trees are:
  o    (oo)      (ooo)
  (o)  (o(o))    (o(oo))
       ((o)(o))  (oo(o))
                 ((o)(oo))
                 (o(o)(o))
                 (o(o(o)))
                 ((o)(o)(o))
                 ((o)(o(o)))
                 (o((o)(o)))
                 ((o)((o)(o)))
(End)

			For n=2, the 2*a(2) = 6 elements are: A+A, A+B, B+B, A*A, A*B, B*B. - _Michael Somos_, Aug 07 2017

Andrew Howroyd, Table of n, a(n) for n = 1..500
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012. - From _N. J. A. Sloane_, Dec 22 2012
N. J. A. Sloane, Transforms
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
Index entries for sequences related to rooted trees

Column 2 of A319254.
Cf. A029856, A031148.
Lone-child-avoiding rooted trees with n leaves are A000669.
Lone-child-avoiding rooted trees with n vertices are A001678.
The locally disjoint case is A331874.
Semi-lone-child-avoiding rooted trees with n vertices are A331934.
Matula-Goebel numbers of these trees are A331935.
Cf. A000081, A005804, A141268, A196545, A291636, A316697, A330465, A331872, A331933, A331964.

terms = 22;
B[x_] = x O[x]^(terms+1);
A[x_] = 1/(1 - x + B[x])^2;
Do[A[x_] = A[x]/(1 - x^k + B[x])^Coefficient[A[x], x, k] + O[x]^(terms+1) // Normal, {k, 2, terms+1}];
Join[{2}, Drop[CoefficientList[A[x], x]/2, 2]] (* Jean-François Alcover, Aug 17 2018, after Michael Somos *)
slaurte[n_]:=If[n==1,{o,{o}},Join@@Table[Union[Sort/@Tuples[slaurte/@ptn]],{ptn,Rest[IntegerPartitions[n]]}]];
Table[Length[slaurte[n]],{n,10}] (* Gus Wiseman, Feb 07 2020 *)

{a(n) = my(A, B); if( n<2, 2*(n>0), B = x * O(x^n); A = 1 / (1 - x + B)^2; for(k=2, n, A /= (1 - x^k + B)^polcoeff(A, k)); polcoeff(A, n)/2)}; /* Michael Somos, Aug 07 2017 */

Doubles (index 2+) under EULER transform.
Product_{k>=1} (1-x^k)^-a(k) = 1 + a(1)*x + Sum_{k>=2} 2*a(k)*x^k. - Michael Somos, Aug 07 2017
a(n) ~ c * d^n / n^(3/2), where d = 6.158893517087396289837838459951206775682824030495453326610366016992093939... and c = 0.1914250508201011360729769525164141605187995730026600722369002... - Vaclav Kotesovec, Aug 17 2018

A108919 Number of series-reduced labeled trees with n nodes.

Original entry on oeis.org

1, 0, 1, 1, 13, 51, 601, 4803, 63673, 775351, 12186061, 196158183, 3661759333, 72413918019, 1583407093633, 36916485570331, 929770285841137, 24904721121298671, 711342228666833173, 21502519995056598639, 687345492498807434461, 23135454269839313430715, 818568166383797223246601, 30357965273255025673685091

Offset: 1

Vladeta Jovovic, Jul 20 2005

"Series-reduced" means that if the tree is rooted at 1, then there is no node with just a single child.

Callan points out that A002792 is an incorrect version of this sequence. - Joerg Arndt, Jul 01 2014

Seiichi Manyama, Table of n, a(n) for n = 1..415
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], 2014.

The rooted version is A060356.

Cf. A000311, A000669, A001678, A292504, A316651, A316652, A318231, A318813, A330465, A330624, A352410.

f[n_] := Sum[(-1)^(n-k)*n!/k!*Binomial[n-1, k-1]*k^(k-1), {k, n}]/n; Table[ f[n], {n, 20}] (* Robert G. Wilson v, Jul 21 2005 *)

a(n) = { 1/n * sum(k=1, n, (-1)^(n-k) * binomial(n,k) * (n-1)!/(k-1)! * k^(k-1) ); } \\ Joerg Arndt, Aug 28 2014

a(n) = A060356(n)/n.
1 = Sum_{n>=0} a(n+1)*(exp(x)-x)^(-n-1)*x^n/n!.
E.g.f.: A(x) = Sum_{n>=0} a(n+1)*x^n/n! satisfies A(x) = exp(x*A(x))/(1+x). - Olivier Gérard, Dec 31 2013 (edited by Gus Wiseman, Dec 31 2019)
E.g.f.: -Integral (LambertW(-x/(1 + x))/x) dx. - Ilya Gutkovskiy, Jul 01 2020

More terms from Robert G. Wilson v, Jul 21 2005
New name (from A002792) by Joerg Arndt, Aug 28 2014
Offset corrected by Gus Wiseman, Dec 31 2019

A330465 Number of non-isomorphic series-reduced rooted trees whose leaves are multisets with a total of n elements.

Original entry on oeis.org

1, 4, 14, 87, 608, 5573, 57876, 687938, 9058892, 130851823, 2048654450, 34488422057, 620046639452, 11839393796270, 238984150459124, 5079583100918338, 113299159314626360, 2644085918303683758, 64393240540265515110, 1632731130253043991252, 43013015553755764179000

Offset: 1

Gus Wiseman, Dec 21 2019

nonn

Also inequivalent leaf-colorings of phylogenetic rooted trees with n labels. A phylogenetic rooted tree is a series-reduced rooted tree whose leaves are (usually disjoint) sets.

			Non-isomorphic representatives of the a(3) = 14 trees:
  ((1)((1)(1)))  ((1)((1)(2)))  ((1)((2)(3)))  ((2)((1)(1)))
  ((1)(1)(1))    ((1)(1)(2))    ((1)(2)(3))    ((2)(1,1))
  ((1)(1,1))     ((1)(1,2))     ((1)(2,3))
  (1,1,1)        (1,1,2)        (1,2,3)

The version where leaves are atoms is A318231.
The case with sets as leaves is A330624.
The case with disjoint sets as leaves is A141268.
Labeled versions are A330467 (strongly normal) and A330469 (normal).
The singleton-reduced version is A330470.
Cf. A000311, A000669, A004114, A005804, A007716, A281118, A289501, A292504, A316651, A316652, A318812, A319312, A330471, A330474, A330625, A339645.

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n), p=sEulerT(x*sv(1) + O(x*x^n))); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sEulerT(x*Ser(v[1..n])), n ) + polcoef(p,n)); x*Ser(v)}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 13 2020

Terms a(7) and beyond from Andrew Howroyd, Dec 13 2020

A331934 Number of semi-lone-child-avoiding rooted trees with n unlabeled vertices.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 15, 29, 62, 129, 279, 602, 1326, 2928, 6544, 14692, 33233, 75512, 172506, 395633, 911108, 2105261, 4880535, 11346694, 26451357, 61813588, 144781303, 339820852, 799168292, 1882845298, 4443543279, 10503486112, 24864797324, 58944602767, 139918663784

Offset: 1

Gus Wiseman, Feb 03 2020

nonn

A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.

			The a(1) = 1 through a(7) = 15 trees:
  o  (o)  (oo)  (ooo)   (oooo)    (ooooo)    (oooooo)
                (o(o))  (o(oo))   (o(ooo))   (o(oooo))
                        (oo(o))   (oo(oo))   (oo(ooo))
                        ((o)(o))  (ooo(o))   (ooo(oo))
                                  ((o)(oo))  (oooo(o))
                                  (o(o)(o))  ((o)(ooo))
                                  (o(o(o)))  ((oo)(oo))
                                             (o(o)(oo))
                                             (o(o(oo)))
                                             (o(oo(o)))
                                             (oo(o)(o))
                                             (oo(o(o)))
                                             ((o)(o)(o))
                                             ((o)(o(o)))
                                             (o((o)(o)))

Andrew Howroyd, Table of n, a(n) for n = 1..1000
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

The same trees counted by leaves are A050381.
The locally disjoint version is A331872.
Matula-Goebel numbers of these trees are A331935.
Lone-child-avoiding rooted trees are A001678.
Cf. A000081, A000669, A198518, A289501, A291636, A306200, A320268, A330465, A330951, A331873, A331874, A331933, A331966.

sse[n_]:=Switch[n,1,{{}},2,{{{}}},_,Join@@Function[c,Union[Sort/@Tuples[sse/@c]]]/@Rest[IntegerPartitions[n-1]]];
Table[Length[sse[n]],{n,10}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=[1,1]); for(n=2, n-1, v=concat(v, EulerT(v)[n] - v[n])); v} \\ Andrew Howroyd, Feb 09 2020

Product_{k > 0} 1/(1 - x^k)^a(k) = A(x) + A(x)/x - x where A(x) = Sum_{k > 0} x^k a(k).

Euler transform is b(1) = 1, b(n > 1) = a(n) + a(n + 1).

Terms a(25) and beyond from Andrew Howroyd, Feb 09 2020

cp's OEIS Frontend

A060356 Expansion of e.g.f.: -LambertW(-x/(1+x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

PARI

Formula

A316651 Number of series-reduced rooted trees with n leaves spanning an initial interval of positive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A006024 Number of labeled mating graphs with n nodes. Also called point-determining graphs.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A006351 Number of series-parallel networks with n labeled edges. Also called yoke-chains by Cayley and MacMahon.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Sage

Formula

A316652 Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A320160 Number of series-reduced balanced rooted trees whose leaves form an integer partition of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs