A330465 -id:A330465 - cp's OEIS frontend

A339645 Triangle read by rows: T(n,k) is the number of inequivalent colorings of lone-child-avoiding rooted trees with n colored leaves using exactly k colors.

1, 1, 1, 2, 3, 2, 5, 17, 12, 5, 12, 73, 95, 44, 12, 33, 369, 721, 512, 168, 33, 90, 1795, 5487, 5480, 2556, 625, 90, 261, 9192, 41945, 58990, 36711, 12306, 2342, 261, 766, 47324, 321951, 625088, 516952, 224241, 57155, 8702, 766, 2312, 249164, 2483192, 6593103, 7141755, 3965673, 1283624, 258887, 32313, 2312

Offset: 1

Andrew Howroyd, Dec 11 2020

Only the leaves are colored. Equivalence is up to permutation of the colors.

Lone-child-avoiding rooted trees are also called planted series-reduced trees in some other sequences.

			Triangle begins:
    1;
    1,     1;
    2,     3,      2;
    5,    17,     12,      5;
   12,    73,     95,     44,     12;
   33,   369,    721,    512,    168,     33;
   90,  1795,   5487,   5480,   2556,    625,    90;
  261,  9192,  41945,  58990,  36711,  12306,  2342,  261;
  766, 47324, 321951, 625088, 516952, 224241, 57155, 8702, 766;
  ...
From _Gus Wiseman_, Jan 02 2021: (Start)
Non-isomorphic representatives of the 39 = 5 + 17 + 12 + 5 trees with four colored leaves:
  (1111)      (1112)      (1123)      (1234)
  (1(111))    (1122)      (1(123))    (1(234))
  (11(11))    (1(112))    (11(23))    (12(34))
  ((11)(11))  (11(12))    (12(13))    ((12)(34))
  (1(1(11)))  (1(122))    (2(113))    (1(2(34)))
              (11(22))    (23(11))
              (12(11))    ((11)(23))
              (12(12))    (1(1(23)))
              (2(111))    ((12)(13))
              ((11)(12))  (1(2(13)))
              (1(1(12)))  (2(1(13)))
              ((11)(22))  (2(3(11)))
              (1(1(22)))
              (1(2(11)))
              ((12)(12))
              (1(2(12)))
              (2(1(11)))
(End)

Andrew Howroyd, PARI Functions for Combinatorial Species, v2, Dec 2020.
Wikipedia, Combinatorial species

The case with only one color is A000669.
Counting by nodes gives A318231.
A labeled version is A319376.
Row sums are A330470.
A000311 counts singleton-reduced phylogenetic trees.
A001678 counts unlabeled lone-child-avoiding rooted trees.
A005121 counts chains of set partitions, with maximal case A002846.
A005804 counts phylogenetic rooted trees with n labels.
A060356 counts labeled lone-child-avoiding rooted trees.
A141268 counts lone-child-avoiding rooted trees with leaves summing to n.
A291636 lists Matula-Goebel numbers of lone-child-avoiding rooted trees.
A316651 counts lone-child-avoiding rooted trees with normal leaves.
A316652 counts lone-child-avoiding rooted trees with strongly normal leaves.
A330465 counts inequivalent leaf-colorings of phylogenetic rooted trees.
Cf. A196545, A213427, A281118, A289501, A292504, A318812, A319312, A330627.

\\ See link above 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)}
{my(A=InequivalentColoringsTriangle(cycleIndexSeries(10))); for(n=1, #A~, print(A[n,1..n]))}

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

Original entry on oeis.org

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

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

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

A330470 Number of non-isomorphic series/singleton-reduced rooted trees on a multiset of size n.

Original entry on oeis.org

1, 1, 2, 7, 39, 236, 1836, 16123, 162008, 1802945, 22012335, 291290460, 4144907830, 62986968311, 1016584428612, 17344929138791, 311618472138440, 5875109147135658, 115894178676866576, 2385755803919949337, 51133201045333895149, 1138659323863266945177, 26296042933904490636133

Offset: 0

Gus Wiseman, Dec 22 2019

nonn

A series/singleton-reduced rooted tree on a multiset m is either the multiset m itself or a sequence of series/singleton-reduced rooted trees, one on each part of a multiset partition of m that is neither minimal (all singletons) nor maximal (only one part).

			Non-isomorphic representatives of the a(4) = 39 trees, with singleton leaves (x) replaced by just x:
  (1111)      (1112)      (1122)      (1123)      (1234)
  (1(111))    (1(112))    (1(122))    (1(123))    (1(234))
  (11(11))    (11(12))    (11(22))    (11(23))    (12(34))
  ((11)(11))  (12(11))    (12(12))    (12(13))    ((12)(34))
  (1(1(11)))  (2(111))    ((11)(22))  (2(113))    (1(2(34)))
              ((11)(12))  (1(1(22)))  (23(11))
              (1(1(12)))  ((12)(12))  ((11)(23))
              (1(2(11)))  (1(2(12)))  (1(1(23)))
              (2(1(11)))              ((12)(13))
                                      (1(2(13)))
                                      (2(1(13)))
                                      (2(3(11)))

The case with all atoms equal or all atoms different is A000669.
Not requiring singleton-reduction gives A330465.
Labeled versions are A316651 (normal orderless) and A330471 (strongly normal).
The case where the leaves are sets is A330626.
Row sums of A339645.
Cf. A000311, A005121, A005804, A141268, A213427, A292504, A292505, A318812, A318848, A318849, A330467, A330469, A330474, A330624.

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sEulerT(x*Ser(v[1..n])), n )); x*Ser(v)}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 11 2020

Terms a(7) and beyond from Andrew Howroyd, Dec 11 2020

A331679 Number of lone-child-avoiding locally disjoint rooted trees whose leaves are positive integers summing to n, with no two distinct leaves directly under the same vertex.

Original entry on oeis.org

1, 2, 3, 8, 16, 48, 116, 341, 928, 2753, 7996, 24254, 73325, 226471, 702122

Offset: 1

Gus Wiseman, Jan 25 2020

A tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex. It is lone-child-avoiding if there are no unary branchings.

			The a(1) = 1 through a(5) = 16 trees:
  1  2     3        4           5
     (11)  (111)    (22)        (11111)
           (1(11))  (1111)      ((11)3)
                    (2(11))     (1(22))
                    (1(111))    (2(111))
                    (11(11))    (1(1111))
                    ((11)(11))  (11(111))
                    (1(1(11)))  (111(11))
                                (1(2(11)))
                                (2(1(11)))
                                (1(1(111)))
                                (1(11)(11))
                                (1(11(11)))
                                (11(1(11)))
                                (1((11)(11)))
                                (1(1(1(11))))

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 non-locally disjoint version is A141268.
Locally disjoint trees counted by vertices are A316473.
The case where all leaves are 1's is A316697.
Number of trees counted by A331678 with all atoms equal to 1.
Matula-Goebel numbers of locally disjoint rooted trees are A316495.
Unlabeled lone-child-avoiding locally disjoint rooted trees are A331680.
Cf. A000081, A000669, A001678, A005804, A060356, A300660, A316471, A316694, A316696, A319312, A330465, A331681.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
usot[n_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[usot/@ptn]],disjointQ[DeleteCases[#,_?AtomQ]]&&SameQ@@Select[#,AtomQ]&],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}],n];
Table[Length[usot[n]],{n,12}]

A331680 Number of lone-child-avoiding locally disjoint unlabeled rooted trees with n vertices.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 9, 16, 26, 45, 72, 124, 201, 341, 561, 947, 1571, 2651, 4434, 7496, 12631, 21423, 36332, 61910, 105641, 180924, 310548, 534713, 923047

Offset: 1

Gus Wiseman, Jan 25 2020

First differs from A320268 at a(11) = 45, A320268(11) = 44.

A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex. Lone-child-avoiding means there are no unary branchings.

			The a(1) = 1 through a(9) = 16 trees (empty column indicated by dot):
  o  .  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)    (oooooooo)
                     (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))   (o(oooooo))
                              (oo(oo))  (oo(ooo))   (oo(oooo))   (oo(ooooo))
                                        (ooo(oo))   (ooo(ooo))   (ooo(oooo))
                                        ((oo)(oo))  (oooo(oo))   (oooo(ooo))
                                        (o(o(oo)))  (o(o(ooo)))  (ooooo(oo))
                                                    (o(oo)(oo))  ((ooo)(ooo))
                                                    (o(oo(oo)))  (o(o(oooo)))
                                                    (oo(o(oo)))  (o(oo(ooo)))
                                                                 (o(ooo(oo)))
                                                                 (oo(o(ooo)))
                                                                 (oo(oo)(oo))
                                                                 (oo(oo(oo)))
                                                                 (ooo(o(oo)))
                                                                 (o((oo)(oo)))
                                                                 (o(o(o(oo))))

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 enriched version is A316696.
The Matula-Goebel numbers of these trees are A331871.
The non-locally disjoint version is A001678.
These trees counted by number of leaves are A316697.
The semi-lone-child-avoiding version is A331872.
Cf. A000081, A000669, A005804, A060356, A141268, A300660, A316471, A316473, A316694, A316495, A319312, A330465, A331679, A331681, A331683.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
strut[n_]:=If[n==1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@Rest[IntegerPartitions[n-1]],disjointQ]];
Table[Length[strut[n]],{n,10}]

A330467 Number of series-reduced rooted trees whose leaves are multisets whose multiset union is a strongly normal multiset of size n.

Original entry on oeis.org

1, 1, 4, 18, 154, 1614, 23733, 396190, 8066984, 183930948, 4811382339, 138718632336, 4451963556127, 155416836338920, 5920554613563841, 242873491536944706, 10725017764009207613, 505671090907469848248, 25415190929321149684700, 1354279188424092012064226

Offset: 0

Gus Wiseman, Dec 22 2019

nonn

A multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.

Also the number of different colorings of phylogenetic trees with n labels using strongly normal multisets of colors. A phylogenetic tree is a series-reduced rooted tree whose leaves are (usually disjoint) sets.

			The a(3) = 18 trees:
  {1,1,1}          {1,1,2}          {1,2,3}
  {{1},{1,1}}      {{1},{1,2}}      {{1},{2,3}}
  {{1},{1},{1}}    {{2},{1,1}}      {{2},{1,3}}
  {{1},{{1},{1}}}  {{1},{1},{2}}    {{3},{1,2}}
                   {{1},{{1},{2}}}  {{1},{2},{3}}
                   {{2},{{1},{1}}}  {{1},{{2},{3}}}
                                    {{2},{{1},{3}}}
                                    {{3},{{1},{2}}}

The singleton-reduced version is A316652.
The unlabeled version is A330465.
Not requiring weakly decreasing multiplicities gives A330469.
The case where the leaves are sets is A330625.
Cf. A000311, A000669, A004114, A005121, A005804, A141268, A292504, A292505, A318812, A318849, A319312, A330471, A330475.

strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
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]]]];
multing[t_,n_]:=Array[(t+#-1)/#&,n,1,Times];
amemo[m_]:=amemo[m]=1+Sum[Product[multing[amemo[s[[1]]],Length[s]],{s,Split[c]}],{c,Select[mps[m],Length[#]>1&]}];
Table[Sum[amemo[m],{m,strnorm[n]}],{n,0,5}]

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n), p=sExp(x*sv(1) + O(x*x^n))); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sExp(x*Ser(v[1..n])), n ) + polcoef(p, n)); 1 + x*Ser(v)}
StronglyNormalLabelingsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 28 2020

Terms a(10) and beyond from Andrew Howroyd, Dec 28 2020

A330469 Number of series-reduced rooted trees whose leaves are multisets with a total of n elements covering an initial interval of positive integers.

Original entry on oeis.org

1, 1, 4, 24, 250, 3744, 73408, 1768088, 50468854, 1664844040, 62304622320, 2607765903568, 120696071556230, 6120415124163512, 337440974546042416, 20096905939846645064, 1285779618228281270718, 87947859243850506008984, 6404472598196204610148232

Offset: 0

Gus Wiseman, Dec 22 2019

nonn

Also the number of different colorings of phylogenetic trees with n labels using a multiset of colors covering an initial interval of positive integers. A phylogenetic tree is a series-reduced rooted tree whose leaves are (usually disjoint) sets.

			The a(3) = 24 trees:
  (123)          (122)          (112)          (111)
  ((1)(23))      ((1)(22))      ((1)(12))      ((1)(11))
  ((2)(13))      ((2)(12))      ((2)(11))      ((1)(1)(1))
  ((3)(12))      ((1)(2)(2))    ((1)(1)(2))    ((1)((1)(1)))
  ((1)(2)(3))    ((1)((2)(2)))  ((1)((1)(2)))
  ((1)((2)(3)))  ((2)((1)(2)))  ((2)((1)(1)))
  ((2)((1)(3)))
  ((3)((1)(2)))

Andrew Howroyd, Table of n, a(n) for n = 0..200

The singleton-reduced version is A316651.
The unlabeled version is A330465.
The strongly normal case is A330467.
The case when leaves are sets is A330764.
Row sums of A330762.
Cf. A000311, A000669, A004114, A005121, A005804, A141268, A292504, A292505, A316652, A318812, A318849, A319312, A330625.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
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]]]];
multing[t_,n_]:=Array[(t+#-1)/#&,n,1,Times];
amemo[m_]:=amemo[m]=1+Sum[Product[multing[amemo[s[[1]]],Length[s]],{s,Split[c]}],{c,Select[mps[m],Length[#]>1&]}];
Table[Sum[amemo[m],{m,allnorm[n]}],{n,0,5}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[]); for(n=1, n, v=concat(v, EulerT(concat(v, [binomial(n+k-1, k-1)]))[n])); v}
seq(n)={concat([1], sum(k=1, n, R(n,k)*sum(r=k, n, binomial(r,k)*(-1)^(r-k))))} \\ Andrew Howroyd, Dec 29 2019

Terms a(9) and beyond from Andrew Howroyd, Dec 29 2019

cp's OEIS Frontend

A339645 Triangle read by rows: T(n,k) is the number of inequivalent colorings of lone-child-avoiding rooted trees with n colored leaves using exactly k colors.

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

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

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

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

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A330470 Number of non-isomorphic series/singleton-reduced rooted trees on a multiset of size n.

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A331679 Number of lone-child-avoiding locally disjoint rooted trees whose leaves are positive integers summing to n, with no two distinct leaves directly under the same vertex.

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