A001678 -id:A001678 - cp's OEIS frontend

A318231 Number of inequivalent leaf-colorings of series-reduced rooted trees with n nodes.

1, 0, 2, 3, 9, 23, 73, 229, 796, 2891, 11118, 44695, 187825, 820320, 3716501, 17413308, 84209071, 419461933, 2148673503, 11301526295, 60956491070, 336744177291, 1903317319015, 10995856040076, 64873456288903, 390544727861462, 2397255454976268, 14993279955728851

Offset: 1

Gus Wiseman, Aug 21 2018

nonn

In a series-reduced rooted tree, every non-leaf node has at least two branches.

			Inequivalent representatives of the a(6) = 23 leaf-colorings:
  (11(11))  (1(111))  (11111)
  (11(12))  (1(112))  (11112)
  (11(22))  (1(122))  (11122)
  (11(23))  (1(123))  (11123)
  (12(11))  (1(222))  (11223)
  (12(12))  (1(223))  (11234)
  (12(13))  (1(234))  (12345)
  (12(33))
  (12(34))

Cf. A000081, A001190, A001678, A003238, A004111, A290689, A291636, A304486.

Cf. A318226, A318227, A318228, A318229, A318230, A318234, A339645, A339648.

\\ 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(concat(v[1..n-2], [0]))), n-1 )); x*Ser(v)}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 11 2020

Terms a(8) 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}]

A331681 One, two, and all numbers of the form 2^k * prime(j) where k > 0 and j already belongs to the sequence.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 16, 24, 26, 28, 32, 38, 48, 52, 56, 64, 74, 76, 86, 96, 104, 106, 112, 128, 148, 152, 172, 178, 192, 202, 208, 212, 214, 224, 256, 262, 296, 304, 326, 344, 356, 384, 404, 416, 424, 428, 446, 448, 478, 512, 524, 526, 592, 608, 622, 652

Offset: 1

Gus Wiseman, Jan 26 2020

nonn

Also Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted semi-identity trees. A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf. Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex. In a semi-identity tree, all non-leaf branches of any given vertex are distinct. Note that these conditions together imply that there is at most one non-leaf branch under any given vertex.
Also Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one non-leaf branch under any given vertex.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches (of the root), which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The sequence of all semi-lone-child-avoiding rooted trees with at most one non-leaf branch under any given vertex, together with their Matula-Goebel numbers, begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  32: (ooooo)
  38: (o(ooo))
  48: (oooo(o))
  52: (oo(o(o)))
  56: (ooo(oo))
  64: (oooooo)
  74: (o(oo(o)))
  76: (oo(ooo))
  86: (o(o(oo)))

Robert Israel, Table of n, a(n) for n = 1..4000

The enumeration of these trees by nodes is A324969 (essentially A000045).
The enumeration of these trees by leaves appears to be A090129(n + 1).
The (non-semi) lone-child-avoiding version is A331683.
Matula-Goebel numbers of rooted semi-identity trees are A306202.
Lone-child-avoiding locally disjoint rooted trees by leaves are A316697.
The set S of numbers with at most one prime index in S is A331784.
Matula-Goebel numbers of locally disjoint rooted trees are A316495.
Cf. A000081, A000669, A001678, A007097, A061775, A196050, A291636, A316470, A316473, A331679, A331680, A331682, A331687, A331783, A331785, A331873.

N:= 1000: # for terms <= N
S:= {1,2}:
with(queue):
Q:= new(1,2):
while not empty(Q) do
  r:= dequeue(Q);
  p:= ithprime(r);
  newS:= {seq(2^i*p,i=1..ilog2(N/p))} minus S;
  S:= S union newS;
  for s in newS do enqueue(Q,s) od:
od:
sort(convert(S,list)); # Robert Israel, Feb 05 2020

uryQ[n_]:=n==1||MatchQ[FactorInteger[n],({{2,},{p,1}}/;uryQ[PrimePi[p]])|{{2,_}}];
Select[Range[100],uryQ]

Intersection of A306202 (semi-identity), A316495 (locally disjoint), and A331935 (semi-lone-child-avoiding). - Gus Wiseman, Jun 09 2020

A331964 Number of semi-lone-child-avoiding rooted identity trees with n vertices.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 2, 2, 4, 6, 10, 16, 27, 44, 74, 123, 209, 353, 602, 1026, 1760, 3019, 5203, 8977, 15538, 26930, 46792, 81415, 141939, 247795, 433307, 758672, 1330219, 2335086, 4104064, 7220937, 12718694, 22424283, 39574443, 69903759, 123584852, 218668323

Offset: 1

Gus Wiseman, Feb 04 2020

nonn

A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf. It is an identity tree if the branches of any given vertex are all distinct.

			The a(9) = 2 through a(12) = 10 semi-lone-child-avoiding rooted identity trees:
  ((o)(o(o(o))))  (o(o)(o(o(o))))   ((o)(o(o)(o(o))))  (o(o)(o(o)(o(o))))
  (o((o)(o(o))))  (o(o(o)(o(o))))   ((o)(o(o(o(o)))))  (o(o)(o(o(o(o)))))
                  (o(o(o(o(o)))))   ((o(o))(o(o(o))))  (o(o(o))(o(o(o))))
                  ((o)((o)(o(o))))  (o((o)(o(o(o)))))  (o(o(o)(o(o(o)))))
                                    (o(o)((o)(o(o))))  (o(o(o(o)(o(o)))))
                                    (o(o((o)(o(o)))))  (o(o(o(o(o(o))))))
                                                       ((o)((o)(o(o(o)))))
                                                       ((o)(o((o)(o(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.
Gus Wiseman, The semi-lone-child-avoiding rooted identity trees with up to 13 vertices.

The non-semi version is A000007.
Matula-Goebel numbers of these trees are A331963.
Rooted identity trees are A004111.
Semi-lone-child-avoiding rooted trees are A331934.
Cf. A000081, A001678, A050381, A276625, A300660, A306200, A316694, A331872, A331875, A331933, A331935, A331936, A331937, A331966.

ssei[n_]:=Switch[n,1,{{}},2,{{{}}},_,Join@@Function[c,Select[Union[Sort/@Tuples[ssei/@c]],UnsameQ@@#&]]/@Rest[IntegerPartitions[n-1]]];
Table[Length[ssei[n]],{n,15}]

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

Terms a(36) and beyond from Andrew Howroyd, Feb 09 2020

A330655 Number of balanced reduced multisystems of weight n whose atoms cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 2, 12, 138, 2652, 78106, 3256404, 182463296, 13219108288, 1202200963522, 134070195402644, 17989233145940910, 2858771262108762492, 530972857546678902490, 113965195745030648131036, 27991663753030583516229824, 7800669355870672032684666900, 2448021231611414334414904013956

Offset: 0

Gus Wiseman, Dec 27 2019

nonn

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

			The a(0) = 1 through a(3) = 12 multisystems:
  {}  {1}  {1,1}  {1,1,1}
           {1,2}  {1,1,2}
                  {1,2,2}
                  {1,2,3}
                  {{1},{1,1}}
                  {{1},{1,2}}
                  {{1},{2,2}}
                  {{1},{2,3}}
                  {{2},{1,1}}
                  {{2},{1,2}}
                  {{2},{1,3}}
                  {{3},{1,2}}

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

Row sums of A330776.
The unlabeled version is A330474.
The strongly normal case is A330475.
The tree version is A330654.
The maximum-depth case is A330676.
The case where the atoms are all different is A005121.
The case where the atoms are all equal is A318813.
Multiset partitions of normal multisets are A255906.
Series-reduced rooted trees with normal leaves are A316651.
Cf. A000311, A000669, A001678, A213427, A318812, A330675, A330679.

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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n,k)={my(v=vector(n), u=vector(n)); v[1]=k; for(n=1, #v, u += v*sum(j=n, #v, (-1)^(j-n)*binomial(j-1,n-1)); v=EulerT(v)); u}
seq(n)={concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k))))} \\ Andrew Howroyd, Dec 30 2019

Terms a(7) and beyond from Andrew Howroyd, Dec 30 2019

A331872 Number of semi-lone-child-avoiding locally disjoint rooted trees with n vertices.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 12, 19, 35, 59, 104, 179, 318, 556, 993, 1772, 3202, 5807, 10643, 19594, 36380, 67915

Offset: 1

Gus Wiseman, Feb 02 2020

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

Locally disjoint means no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex.

			The a(1) = 1 through a(8) = 19 trees:
  o  (o)  (oo)  (ooo)   (oooo)    (ooooo)    (oooooo)     (ooooooo)
                (o(o))  (o(oo))   (o(ooo))   (o(oooo))    (o(ooooo))
                        (oo(o))   (oo(oo))   (oo(ooo))    (oo(oooo))
                        ((o)(o))  (ooo(o))   (ooo(oo))    (ooo(ooo))
                                  (o(o)(o))  (oooo(o))    (oooo(oo))
                                  (o(o(o)))  ((oo)(oo))   (ooooo(o))
                                             (o(o(oo)))   (o(o(ooo)))
                                             (o(oo(o)))   (o(oo)(oo))
                                             (oo(o)(o))   (o(oo(oo)))
                                             (oo(o(o)))   (o(ooo(o)))
                                             ((o)(o)(o))  (oo(o(oo)))
                                             (o((o)(o)))  (oo(oo(o)))
                                                          (ooo(o)(o))
                                                          (ooo(o(o)))
                                                          (o(o)(o)(o))
                                                          (o(o(o)(o)))
                                                          (o(o(o(o))))
                                                          (oo((o)(o)))
                                                          ((o)((o)(o)))

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.

Not requiring lone-child-avoidance gives A316473.
The non-semi version is A331680.
The Matula-Goebel numbers of these trees are A331873.
The same trees counted by number of leaves are A331874.
Not requiring local disjointness gives A331934.
Lone-child-avoiding rooted trees are A001678.
Cf. A000081, A050381, A316696, A316697, A331678, A331679, A331681, A331686, A331687, A331871, A331935.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
strutsemi[n_]:=If[n==1,{{}},If[n==2,{{{}}},Select[Join@@Function[c,Union[Sort/@Tuples[strutsemi/@c]]]/@Rest[IntegerPartitions[n-1]],disjointQ]]];
Table[Length[strutsemi[n]],{n,8}]

A316653 Number of series-reduced rooted identity trees with n leaves spanning an initial interval of positive integers.

Original entry on oeis.org

1, 1, 6, 58, 774, 13171, 272700, 6655962, 187172762, 5959665653, 211947272186, 8327259067439, 358211528524432, 16744766791743136, 845195057333580332, 45814333121920927067, 2654330505021077873594, 163687811930206581162063, 10705203621191765328300832

Offset: 1

Gus Wiseman, Jul 09 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is an identity tree if no branch appears multiple times under the same root.

			The a(3) = 6 trees are (1(12)), (2(12)), (1(23)), (2(13)), (3(12)), (123).

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

Cf. A000081, A000311, A000669, A001678, A004111, A005804, A034691, A141268, A292504, A300660.

Cf. A316651, A316652, 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,Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],UnsameQ@@#&]];
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}]

\\ here R(n,2) is A031148.
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
R(n,k)={my(v=[k]); for(n=2, n, v=concat(v, WeighT(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

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

A316656 Number of series-reduced rooted identity 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, 0, 1, 0, 1, 0, 4, 3, 1, 0, 9, 0, 1, 6, 26, 0, 36, 0, 16, 10, 1, 0, 92, 21, 1, 197, 25, 0, 100, 0, 236, 15, 1, 53, 474

Offset: 1

Gus Wiseman, Jul 09 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is an identity tree if no branch appears multiple times under the same root.

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:
   4: (12)
   5:
   6: (1(12))
   7:
   8: (1(23)), (2(13)), (3(12)), (123)
   9: (1(2(12))), (2(1(12))), (12(12))
  10: (1(1(12)))
  11:
  12: (1(1(23))), (1(2(13))), (1(3(12))), (1(123)), (2(1(13))), (3(1(12))), ((12)(13)), (12(13)), (13(12))

Cf. A000081, A000311, A000669, A001678, A004111, A005804, A056239, A141268, A181821, A292504, A296150, A300660, A304660.

Cf. A316651, A316652, A316653. A316654, A316655.

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,Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],UnsameQ@@#&]];
Table[Length[gro[Flatten[MapIndexed[Table[#2,{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]]],{n,30}]

a(prime(n>1)) = 0.

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

A330675 Number of balanced reduced multisystems of maximum depth whose atoms constitute a strongly normal multiset of size n.

Original entry on oeis.org

1, 1, 2, 6, 43, 440, 7158, 151414

Offset: 0

Gus Wiseman, Dec 30 2019

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

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

			The a(2) = 2 and a(3) = 6 multisystems:
  {1,1}  {{1},{1,1}}
  {1,2}  {{1},{1,2}}
         {{1},{2,3}}
         {{2},{1,1}}
         {{2},{1,3}}
         {{3},{1,2}}
The a(4) = 43 multisystems (commas and outer brackets elided):
  {{1}}{{1}{11}} {{1}}{{1}{12}} {{1}}{{1}{22}} {{1}}{{1}{23}} {{1}}{{2}{34}}
  {{11}}{{1}{1}} {{11}}{{1}{2}} {{11}}{{2}{2}} {{11}}{{2}{3}} {{12}}{{3}{4}}
                 {{1}}{{2}{11}} {{1}}{{2}{12}} {{1}}{{2}{13}} {{1}}{{3}{24}}
                 {{12}}{{1}{1}} {{12}}{{1}{2}} {{12}}{{1}{3}} {{13}}{{2}{4}}
                 {{2}}{{1}{11}} {{2}}{{1}{12}} {{1}}{{3}{12}} {{1}}{{4}{23}}
                                {{2}}{{2}{11}} {{13}}{{1}{2}} {{14}}{{2}{3}}
                                {{22}}{{1}{1}} {{2}}{{1}{13}} {{2}}{{1}{34}}
                                               {{2}}{{3}{11}} {{2}}{{3}{14}}
                                               {{23}}{{1}{1}} {{23}}{{1}{4}}
                                               {{3}}{{1}{12}} {{2}}{{4}{13}}
                                               {{3}}{{2}{11}} {{24}}{{1}{3}}
                                                              {{3}}{{1}{24}}
                                                              {{3}}{{2}{14}}
                                                              {{3}}{{4}{12}}
                                                              {{34}}{{1}{2}}
                                                              {{4}}{{1}{23}}
                                                              {{4}}{{2}{13}}
                                                              {{4}}{{3}{12}}

The case with all atoms equal is A000111.
The case with all atoms different is A006472.
The version allowing all depths is A330475.
The unlabeled version is A330663.
The version where the atoms are the prime indices of n is A330665.
The (weakly) normal version is A330676.
The version where the degrees are the prime indices of n is A330728.
Multiset partitions of strongly normal multisets are A035310.
Series-reduced rooted trees with strongly normal leaves are A316652.
Cf. A000311, A000669, A001055, A001678, A005121, A005804, A316651, A318812, A330467, A330474, A330625, A330628, A330664, A330677, A330679.

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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

cp's OEIS Frontend

A318231 Number of inequivalent leaf-colorings of series-reduced rooted trees with n nodes.

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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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Mathematica

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

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Mathematica

A331681 One, two, and all numbers of the form 2^k * prime(j) where k > 0 and j already belongs to the sequence.

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Maple

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A331964 Number of semi-lone-child-avoiding rooted identity trees with n vertices.

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Mathematica

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A330655 Number of balanced reduced multisystems of weight n whose atoms cover an initial interval of positive integers.

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Mathematica

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A331872 Number of semi-lone-child-avoiding locally disjoint rooted trees with n vertices.

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Mathematica

A316653 Number of series-reduced rooted identity trees with n leaves spanning an initial interval of positive integers.

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