A141268 -id:A141268 - cp's OEIS frontend

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

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

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

A007713 Number of 4-level rooted trees with n leaves.

Original entry on oeis.org

1, 1, 4, 10, 30, 75, 206, 518, 1344, 3357, 8429, 20759, 51044, 123973, 299848, 719197, 1716563, 4070800, 9607797, 22555988, 52718749, 122655485, 284207304, 655894527, 1508046031, 3454808143, 7887768997, 17949709753, 40719611684, 92096461012, 207697731344

Offset: 0

N. J. A. Sloane

			From _Gus Wiseman_, Oct 11 2018: (Start)
Also the number of multiset partitions of multiset partitions of integer partitions of n. For example, the a(1) = 1 through a(4) = 30 multiset partitions are:
  ((1))  ((2))       ((3))            ((4))
         ((11))      ((12))           ((13))
         ((1)(1))    ((111))          ((22))
         ((1))((1))  ((1)(2))         ((112))
                     ((1)(11))        ((1111))
                     ((1))((2))       ((1)(3))
                     ((1))((11))      ((2)(2))
                     ((1)(1)(1))      ((1)(12))
                     ((1))((1)(1))    ((2)(11))
                     ((1))((1))((1))  ((1)(111))
                                      ((11)(11))
                                      ((1))((3))
                                      ((2))((2))
                                      ((1))((12))
                                      ((1)(1)(2))
                                      ((2))((11))
                                      ((1))((111))
                                      ((1)(1)(11))
                                      ((11))((11))
                                      ((1))((1)(2))
                                      ((2))((1)(1))
                                      ((1))((1)(11))
                                      ((1)(1)(1)(1))
                                      ((11))((1)(1))
                                      ((1))((1))((2))
                                      ((1))((1))((11))
                                      ((1))((1)(1)(1))
                                      ((1)(1))((1)(1))
                                      ((1))((1))((1)(1))
                                      ((1))((1))((1))((1))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
B. A. Huberman and T. Hogg, Complexity and adaptation, Evolution, games and learning (Los Alamos, N.M., 1985). Phys. D 22 (1986), no. 1-3, 376-384.
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Column k=4 of A290353.
Main diagonal of A055886.
Cf. A001970, A047968, A050342, A089259, A141268, A258466, A261049, A319066, A320328, A320330, A320331.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: b0:= etr(1): b1:= etr(b0): a:= etr(b1): seq(a(n), n=0..30); # Alois P. Heinz, Sep 08 2008

i[ n_, m_ ] := 1 /; m==1 || n==0; i[ n_, m_ ] := (i[ n, m ]=1/n Sum[ i[ k, m ] Plus @@ ((# i[ #, m-1 ])& /@ Divisors[ n-k ]), {k, 0, n-1} ]) /; n>0 && m>1
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; b0 = etr[Function[1]]; b1 = etr[b0]; a = etr[b1]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

Euler transform applied thrice to all-1's sequence.

A320154 Number of series-reduced balanced rooted trees whose leaves form a set partition of {1,...,n}.

Original entry on oeis.org

1, 2, 5, 18, 92, 588, 4328, 35920, 338437, 3654751, 45105744, 625582147, 9539374171, 157031052142, 2757275781918, 51293875591794, 1007329489077804, 20840741773898303, 453654220906310222, 10380640686263467204, 249559854371799622350, 6301679967177242849680

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 phylogenetic rooted trees on n distinct labels.

			The a(1) = 1 through a(4) = 18 rooted trees:
  (1)  (12)      (123)        (1234)
       ((1)(2))  ((1)(23))    ((1)(234))
                 ((2)(13))    ((12)(34))
                 ((3)(12))    ((13)(24))
                 ((1)(2)(3))  ((14)(23))
                              ((2)(134))
                              ((3)(124))
                              ((4)(123))
                              ((1)(2)(34))
                              ((1)(3)(24))
                              ((1)(4)(23))
                              ((2)(3)(14))
                              ((2)(4)(13))
                              ((3)(4)(12))
                              ((1)(2)(3)(4))
                              (((1)(2))((3)(4)))
                              (((1)(3))((2)(4)))
                              (((1)(4))((2)(3)))

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

Cf. A000081, A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A292504, A300660, A319312.

Cf. A320155, A320160, A316624, A320169, A320173, A320176, A320179.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
gug[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[gug/@mtn]],{mtn,Select[sps[m],Length[#]>1&]}],m];
Table[Length[Select[gug[Range[n]],SameQ@@Length/@Position[#,_Integer]&]],{n,9}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n,k)={my(u=vector(n), v=vector(n)); u[1]=k; u=EulerT(u); while(u, v+=u; u=EulerT(u)-u); v}
seq(n)={my(M=Mat(vectorv(n,k,b(n,k)))); vector(n, k, sum(i=1, k, binomial(k,i)*(-1)^(k-i)*M[i,k]))} \\ Andrew Howroyd, Oct 26 2018

Terms a(9) and beyond from Andrew Howroyd, Oct 26 2018

A316655 Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 5, 4, 12, 9, 12, 17, 33, 29, 44, 26, 90, 90, 261, 68, 168, 93, 766, 144, 197, 307, 575, 269, 2312, 428, 7068, 236, 625, 1017, 863, 954, 21965, 3409, 2342, 712

Offset: 1

Gus Wiseman, Jul 09 2018

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

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of sets of trees begins:
1:
2: 1
3: (11)
4: (12)
5: (1(11)), (111)
6: (1(12)), (2(11)), (112)
7: (1(1(11))), (1(111)), ((11)(11)), (11(11)), (1111)
8: (1(23)), (2(13)), (3(12)), (123)
9: (1(1(22))), (1(2(12))), (1(122)), (2(1(12))), (2(2(11))), (2(112)), ((11)(22)), ((12)(12)), (11(22)), (12(12)), (22(11)), (1122)

Cf. A000081, A000311, A000669, A001678, A005804, A056239, A141268, A181821, A292504, A296150, A300660, A304660.

Cf. A316651, A316652, A316653, A316654, 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[Length[gro[Flatten[MapIndexed[Table[#2,{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]]],{n,20}]

a(prime(n)) = A000669(n).

a(2^n) = A000311(n).

a(37)-a(40) from Robert Price, Sep 13 2018

A316694 Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves form an integer partition of n.

Original entry on oeis.org

1, 1, 2, 3, 6, 13, 28, 62, 143, 338, 804, 1948, 4789, 11886, 29796, 75316, 191702, 491040, 1264926, 3274594, 8514784, 22229481, 58243870

Offset: 1

Gus Wiseman, Jul 10 2018

A rooted tree is lone-child-avoiding if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other (unequal) branch of the same root. It is an identity tree if no branch appears multiple times under the same root.

			The a(7) = 28 rooted trees:
  7,
  (16),
  (25),
  (1(15)),
  (34),
  (1(24)), (2(14)), (4(12)), (124),
  (1(1(14))),
  (3(13)),
  (2(23)),
  (1(1(23))), (1(2(13))), (1(3(12))), (1(123)), (2(1(13))), (3(1(12))), (12(13)), (13(12)),
  (1(1(1(13)))),
  (2(2(12))),
  (1(1(2(12)))), (1(2(1(12)))), (1(12(12))), (2(1(1(12)))), (12(1(12))),
  (1(1(1(1(12))))).
Missing from this list but counted by A300660 are ((12)(13)) and ((12)(1(12))).

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Cf. A000081, A000669, A001678, A141268, A292504, A316653, A316654, A316656.
The semi-identity tree version is A212804.
Not requiring local disjointness gives A300660.
The non-identity tree version is A316696.
This is the case of A331686 where all leaves are singletons.
Rooted identity trees are A004111.
Locally disjoint rooted identity trees are A316471.
Lone-child-avoiding locally disjoint rooted trees are A331680.
Locally disjoint enriched identity p-trees are A331684.
Cf. A306200, A316697, A331678, A331679, A331681, A331683, A331783, A331874.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],And[UnsameQ@@#,disjointQ[#]]&],{ptn,Rest[IntegerPartitions[n]]}],{n}];
Table[Length[nms[n]],{n,10}]

a(21)-a(23) from Robert Price, Sep 16 2018

Updated with corrected terminology by Gus Wiseman, Feb 06 2020

A316697 Number of series-reduced locally disjoint rooted trees with n unlabeled leaves.

Original entry on oeis.org

1, 1, 2, 5, 10, 24, 49, 112, 241, 548, 1218, 2839, 6547, 15572, 37179, 90555, 222065, 552576, 1384820, 3506475, 8936121, 22941280

Offset: 1

Gus Wiseman, Jul 10 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other (unequal) branch of the same root.

			The a(5) = 10 trees:
  (o(o(o(oo))))
  (o(o(ooo)))
  (o((oo)(oo)))
  (o(oo(oo)))
  (o(oooo))
  (oo(o(oo)))
  (oo(ooo))
  (o(oo)(oo))
  (ooo(oo))
  (ooooo)
Missing from this list but counted by A000669 are ((oo)(o(oo))) and ((oo)(ooo)).

Cf. A000081, A000669, A001678, A141268, A316471, A316471, A316475, A316495.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
nms[n_]:=nms[n]=If[n==1,{{1}},Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],disjointQ],{ptn,Rest[IntegerPartitions[n]]}]];
Table[Length[nms[n]],{n,15}]

a(18)-a(22) from Robert Price, Sep 14 2018

A331686 Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 2, 4, 8, 17, 41, 103, 280, 793, 2330, 6979, 21291

Offset: 1

Gus Wiseman, Jan 31 2020

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. In an identity tree, all branches of any given vertex are distinct.

			The a(1) = 1 through a(5) = 17 trees:
  (1)  (2)   (3)       (4)            (5)
       (11)  (12)      (13)           (14)
             (111)     (22)           (23)
             ((1)(2))  (112)          (113)
                       (1111)         (122)
                       ((1)(3))       (1112)
                       ((2)(11))      (11111)
                       ((1)((1)(2)))  ((1)(4))
                                      ((2)(3))
                                      ((1)(22))
                                      ((3)(11))
                                      ((2)(111))
                                      ((1)((1)(3)))
                                      ((2)((1)(2)))
                                      ((11)((1)(2)))
                                      ((1)((2)(11)))
                                      ((1)((1)((1)(2))))

The non-identity version is A331678.
The case where the leaves are all singletons is A316694.
Identity trees are A004111.
Locally disjoint identity trees are A316471.
Locally disjoint enriched identity p-trees are A331684.
Lone-child-avoiding locally disjoint rooted semi-identity trees are A212804.
Cf. A000669, A001678, A005804, A141268, A300660, A316697, A319312, A331679, A331683, A331783, A331874, A331875.

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]]]];
disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
mpti[m_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[mpti/@p]],UnsameQ@@#&&disjointQ[#]&],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[mpti[m]],{m,Sort/@IntegerPartitions[n]}],{n,8}]

A316696 Number of lone-child-avoiding locally disjoint rooted trees whose leaves form an integer partition of n.

Original entry on oeis.org

1, 2, 4, 11, 27, 80, 218, 654, 1923, 5924, 18310, 58176, 186341, 606814, 1993420, 6618160, 22134640

Offset: 1

Gus Wiseman, Jul 10 2018

A rooted tree is lone-child-avoiding if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other (unequal) branch of the same root.

			The a(4) = 11 rooted trees:
  4,
  (13),
  (22),
  (1(12)), (2(11)), (112),
  (1(1(11))), (1(111)), ((11)(11)), (11(11)), (1111).

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Matula-Goebel numbers of locally disjoint rooted trees are A316495.
The case where all leaves are 1's is A316697.
Lone-child-avoiding locally disjoint rooted trees are A331680.
Cf. A000669, A001678, A141268, A316473, A316652, A331678, A331686, A331687, A331871, A331872, A331874.

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

a(16)-a(17) from Robert Price, Sep 16 2018

Terminology corrected by Gus Wiseman, Feb 06 2020

cp's OEIS Frontend

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

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

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

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A007713 Number of 4-level rooted trees with n leaves.

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Maple

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A320154 Number of series-reduced balanced rooted trees whose leaves form a set partition of {1,...,n}.

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A316655 Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.

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A316694 Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves form an integer partition of n.

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