A005804 -id:A005804 - cp's OEIS frontend

A330676 Number of balanced reduced multisystems of weight n and maximum depth whose atoms cover an initial interval of positive integers.

1, 1, 2, 8, 70, 1012, 21944, 665708, 26917492, 1399033348, 90878863352, 7214384973908, 687197223963640, 77354805301801012, 10158257981179981304, 1539156284259756811748, 266517060496258245459352, 52301515332984084095078308, 11546416513975694879642736152

Offset: 0

Gus Wiseman, Dec 30 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.

A finite multiset is normal if it covers an initial interval of positive integers.

			The a(0) = 1 through a(3) = 8 multisystems:
  {}  {1}  {1,1}  {{1},{1,1}}
           {1,2}  {{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 A330778.
The case with all atoms equal is A000111.
The case with all atoms different is A006472.
The version allowing all depths is A330655.
The unlabeled version is A330663.
The version where the atoms are the prime indices of n is A330665.
The strongly normal version is A330675.
The version where the degrees are the prime indices of n is A330728.
Multiset partitions of normal multisets are A255906.
Series-reduced rooted trees with normal leaves are A316651.
Cf. A000669, A001055, A005121, A005804, A318812, A330469, A330474, A330654, A330664, A330677, 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, for(i=n, #v, u[i] += v[i]*(-1)^(i-n)*binomial(i-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 2020

Terms a(8) and beyond from Andrew Howroyd, Dec 30 2019

A331678 Number of lone-child-avoiding locally disjoint rooted trees whose leaves are integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 3, 6, 18, 44, 149, 450, 1573, 5352, 19283, 69483, 257206

Offset: 1

Gus Wiseman, Jan 25 2020

Lone-child-avoiding means there are no unary branchings. Locally disjoint means no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex.

			The a(1) = 1 through a(4) = 18 trees:
  (1)  (2)       (3)            (4)
       (11)      (12)           (13)
       ((1)(1))  (111)          (22)
                 ((1)(2))       (112)
                 ((1)(1)(1))    (1111)
                 ((1)((1)(1)))  ((1)(3))
                                ((2)(2))
                                ((2)(11))
                                ((11)(11))
                                ((1)(1)(2))
                                ((1)((1)(2)))
                                ((2)((1)(1)))
                                ((1)(1)(1)(1))
                                ((11)((1)(1)))
                                ((1)((1)(1)(1)))
                                ((1)(1)((1)(1)))
                                (((1)(1))((1)(1)))
                                ((1)((1)((1)(1))))

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 case where all leaves are singletons is A316696.
The case where all leaves are (1) is A316697.
The non-locally disjoint version is A319312.
The case with all atoms equal to 1 is A331679.
The identity tree case is A331686.
Cf. A000081, A000669, A001678, A005804, A060356, A141268, A196545, A300660, A316471, A316694, A316495, A330465, A331680, A331687.

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]],disjointQ],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[mpti[m]],{m,Sort/@IntegerPartitions[n]}],{n,8}]

A319376 Triangle read by rows: T(n,k) is the number of lone-child-avoiding rooted trees with n leaves of exactly k colors.

Original entry on oeis.org

1, 1, 1, 2, 6, 4, 5, 30, 51, 26, 12, 146, 474, 576, 236, 33, 719, 3950, 8572, 8060, 2752, 90, 3590, 31464, 108416, 175380, 134136, 39208, 261, 18283, 245916, 1262732, 3124650, 4014348, 2584568, 660032, 766, 94648, 1908858, 14047288, 49885320, 95715728, 101799712, 56555904, 12818912

Offset: 1

Andrew Howroyd, Sep 17 2018

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

			Triangle begins:
    1;
    1,     1;
    2,     6,      4;
    5,    30,     51,      26;
   12,   146,    474,     576,     236;
   33,   719,   3950,    8572,    8060,   2752;
   90,  3590,  31464,  108416,  175380,  134136,   39208;
  261, 18283, 245916, 1262732, 3124650, 4014348, 2584568, 660032;
  ...
From _Gus Wiseman_, Dec 31 2020: (Start)
The 12 trees counted by row n = 3:
  (111)    (112)    (123)
  (1(11))  (122)    (1(23))
           (1(12))  (2(13))
           (1(22))  (3(12))
           (2(11))
           (2(12))
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Columns k=1..2 are A000669, A319377.
Main diagonal is A000311.
Row sums are A316651.
Cf. A141610, A242249, A255517, A256064, A256068, A319254, A319541, A326396.
The unlabeled version, counting inequivalent leaf-colorings of lone-child-avoiding rooted trees, is A330465.
Lone-child-avoiding rooted trees are counted by A001678 (shifted left once).
Labeled lone-child-avoiding rooted trees are counted by A060356.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Cf. A000014, A000081, A000169, A005804, A206429, A330951.

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)):
T:= (n, k)-> add(A(n, k-j)*(-1)^j*binomial(k, j), j=0..k-1):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Sep 18 2018

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]];
T[n_, k_] := Sum[(-1)^(k - i)*Binomial[k, i]*A[n, i], {i, 1, k}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 24 2019, after Alois P. Heinz *)
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]]]];
mtot[m_]:=Prepend[Join@@Table[Tuples[mtot/@p],{p,Select[mps[m],1Gus Wiseman, Dec 31 2020 *)

\\ here R(n,k) is k-th column of A319254.
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}
M(n)={my(v=vector(n, k, R(n,k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k,i)*v[i])))}
{my(T=M(10)); for(n=1, #T~, print(T[n, ][1..n]))}

T(n,k) = Sum_{i=1..k} (-1)^(k-i)*binomial(k,i)*A319254(n,i).

Sum_{k=1..n} k * T(n,k) = A326396(n). - Alois P. Heinz, Sep 11 2019

A320155 Number of series-reduced balanced rooted trees with n labeled leaves.

Original entry on oeis.org

1, 1, 1, 4, 11, 41, 162, 1030, 7205, 55522, 442443, 3810852, 35272030, 351697516, 3735838550, 42719792640, 529195988635, 7128835815387, 103651381499810, 1610812109555323, 26489497655582729, 457497408108551450, 8248899117402701046, 154624472715479106919

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.

			The a(1) = 1 through a(5) = 11 rooted trees:
  1  (12)  (123)    (1234)      (12345)
                  ((12)(34))  ((12)(345))
                  ((13)(24))  ((13)(245))
                  ((14)(23))  ((14)(235))
                              ((15)(234))
                              ((23)(145))
                              ((24)(135))
                              ((25)(134))
                              ((34)(125))
                              ((35)(124))
                              ((45)(123))

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

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

Cf. A320154, A320160, A316624, A320169, A320173, A320179.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
phy2[labs_]:=If[Length[labs]==1,labs,Union@@Table[Sort/@Tuples[phy2/@ptn],{ptn,Select[sps[Sort[labs]],Length[#1]>1&]}]];
Table[Length[Select[phy2[Range[n]],SameQ@@Length/@Position[#,_Integer]&]],{n,7}]

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; 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

E.g.f. A(x) satisfies A(x) = x + A(exp(x)-x-1). - Ira M. Gessel, Nov 17 2021

Terms a(10) and beyond from Andrew Howroyd, Oct 26 2018

A320173 Number of inequivalent colorings of series-reduced balanced rooted trees with n leaves.

Original entry on oeis.org

1, 2, 3, 12, 23, 84, 204, 830, 2940, 13397, 58794, 283132, 1377302, 7087164, 37654377, 209943842, 1226495407, 7579549767, 49541194089, 341964495985, 2476907459261, 18703210872343, 146284738788714, 1179199861398539, 9760466433602510, 82758834102114911, 717807201648148643

Offset: 1

Gus Wiseman, Oct 07 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.

			Inequivalent representatives of the a(1) = 1 through a(5) = 23 colorings:
  1  (11)  (111)    (1111)      (11111)
     (12)  (112)    (1112)      (11112)
           (123)    (1122)      (11122)
                    (1123)      (11123)
                    (1234)      (11223)
                  ((11)(11))    (11234)
                  ((11)(12))    (12345)
                  ((11)(22))  ((11)(111))
                  ((11)(23))  ((11)(112))
                  ((12)(12))  ((11)(122))
                  ((12)(13))  ((11)(123))
                  ((12)(34))  ((11)(223))
                              ((11)(234))
                              ((12)(111))
                              ((12)(112))
                              ((12)(113))
                              ((12)(123))
                              ((12)(134))
                              ((12)(345))
                              ((13)(122))
                              ((22)(111))
                              ((23)(111))
                              ((23)(114))

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

Cf. A320154, A320155, A320160, A320174, A320175, A320179, A339645.

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(p=x*sv(1) + O(x*x^n), q=0); while(p, q+=p; p=sEulerT(p)-1-p); q}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 11 2020

Terms a(8) and beyond from Andrew Howroyd, Dec 11 2020

A330471 Number of series/singleton-reduced rooted trees on strongly normal multisets of size n.

Original entry on oeis.org

1, 1, 2, 9, 69, 623, 7803, 110476, 1907428

Offset: 0

Gus Wiseman, Dec 23 2019

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

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). This is a multiset generalization of singleton-reduced phylogenetic trees (A000311).

			The a(0) = 1 through a(3) = 9 trees:
  ()  (1)  (11)  (111)
           (12)  (112)
                 (123)
                 ((1)(11))
                 ((1)(12))
                 ((1)(23))
                 ((2)(11))
                 ((2)(13))
                 ((3)(12))
The a(4) = 69 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))    (13(24))
  (1(1(11)))  (2(111))    (2(112))    (13(12))    (14(23))
              ((11)(12))  (22(11))    (2(113))    (2(134))
              (1(1(12)))  ((11)(22))  (23(11))    (23(14))
              (1(2(11)))  (1(1(22)))  (3(112))    (24(13))
              (2(1(11)))  ((12)(12))  ((11)(23))  (3(124))
                          (1(2(12)))  (1(1(23)))  (34(12))
                          (2(1(12)))  ((12)(13))  (4(123))
                          (2(2(11)))  (1(2(13)))  ((12)(34))
                                      (1(3(12)))  (1(2(34)))
                                      (2(1(13)))  ((13)(24))
                                      (2(3(11)))  (1(3(24)))
                                      (3(1(12)))  ((14)(23))
                                      (3(2(11)))  (1(4(23)))
                                                  (2(1(34)))
                                                  (2(3(14)))
                                                  (2(4(13)))
                                                  (3(1(24)))
                                                  (3(2(14)))
                                                  (3(4(12)))
                                                  (4(1(23)))
                                                  (4(2(13)))
                                                  (4(3(12)))

The case with all atoms different is A000311.
The case with all atoms equal is A196545.
The orderless version is A316652.
The unlabeled version is A330470.
The case where the leaves are sets is A330628.
The version for just normal (not strongly normal) is A330654.
Cf. A000669, A004114, A005121, A005804, A281118, A318812, A318848, A319312, A330465, A330467, A330475.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
mtot[m_]:=Prepend[Join@@Table[Tuples[mtot/@p],{p,Select[mps[m],Length[#]>1&&Length[#]

A330625 Number of series-reduced rooted trees whose leaves are sets (not necessarily disjoint) with multiset union a strongly normal multiset of size n.

Original entry on oeis.org

1, 1, 3, 14, 123, 1330, 19694

Offset: 0

Gus Wiseman, Dec 25 2019

A rooted tree is series-reduced if it has no unary branchings, so every non-leaf node covers at least two other nodes.

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

			The a(1) = 1 through a(3) = 14 trees:
  {1}  {1,2}      {1,2,3}
       {{1},{1}}  {{1},{1,2}}
       {{1},{2}}  {{1},{2,3}}
                  {{2},{1,3}}
                  {{3},{1,2}}
                  {{1},{1},{1}}
                  {{1},{1},{2}}
                  {{1},{2},{3}}
                  {{1},{{1},{1}}}
                  {{1},{{1},{2}}}
                  {{1},{{2},{3}}}
                  {{2},{{1},{1}}}
                  {{2},{{1},{3}}}
                  {{3},{{1},{2}}}

The generalization where the leaves are multisets is A330467.
The singleton-reduced case is A330628.
The unlabeled version is A330624.
The case with all atoms distinct is A005804.
The case with all atoms equal is A196545.
The case where all leaves are singletons is A330471.
Cf. A000669, A004111, A141268, A300660, A316652, A330469, A330475, A330626, A330668, A330675.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
srtrees[m_]:=Prepend[Join@@Table[Tuples[srtrees/@p],{p,Select[mps[m],Length[#1]>1&]}],m];
Table[Sum[Length[Select[srtrees[s],FreeQ[#,{_,x_Integer,x_Integer,_}]&]],{s,strnorm[n]}],{n,0,5}]

A330626 Number of non-isomorphic series/singleton-reduced rooted trees whose leaves are sets (not necessarily disjoint) with a total of n atoms.

Original entry on oeis.org

1, 1, 1, 3, 17, 100, 755

Offset: 0

Gus Wiseman, Dec 26 2019

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(1) = 1 through a(4) = 17 trees:
  {1}  {1,2}  {1,2,3}      {1,2,3,4}
              {{1},{1,2}}  {{1},{1,2,3}}
              {{1},{2,3}}  {{1,2},{1,2}}
                           {{1,2},{1,3}}
                           {{1},{2,3,4}}
                           {{1,2},{3,4}}
                           {{1},{1},{1,2}}
                           {{1},{1},{2,3}}
                           {{1},{2},{1,2}}
                           {{1},{2},{1,3}}
                           {{1},{2},{3,4}}
                           {{1},{{1},{1,2}}}
                           {{1},{{1},{2,3}}}
                           {{1},{{2},{1,2}}}
                           {{1},{{2},{1,3}}}
                           {{1},{{2},{3,4}}}
                           {{2},{{1},{1,3}}}

The non-singleton-reduced version is A330624.
The generalization where leaves are multisets is A330470.
A labeled version is A330628 (strongly normal).
The case with all atoms distinct is A004114.
The balanced version is A330668.
Cf. A000669, A004111, A005804, A007716, A141268, A330465, A330625, A330627, A330654, A330663, A330677.

A330628 Number of series/singleton-reduced rooted trees on strongly normal multisets of size n whose leaves are sets (not necessarily disjoint).

Original entry on oeis.org

1, 1, 1, 5, 42, 423, 5458, 80926

Offset: 0

Gus Wiseman, Dec 26 2019

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).

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

			The a(4) = 42 trees:
  {{1}{1}{12}}    {{12}{12}}      {{1}{123}}      {1234}
  {{1}{{1}{12}}}  {{1}{2}{12}}    {{12}{13}}      {{1}{234}}
                  {{1}{{2}{12}}}  {{1}{1}{23}}    {{12}{34}}
                  {{2}{{1}{12}}}  {{1}{2}{13}}    {{13}{24}}
                                  {{1}{3}{12}}    {{14}{23}}
                                  {{1}{{1}{23}}}  {{2}{134}}
                                  {{1}{{2}{13}}}  {{3}{124}}
                                  {{1}{{3}{12}}}  {{4}{123}}
                                  {{2}{{1}{13}}}  {{1}{2}{34}}
                                  {{3}{{1}{12}}}  {{1}{3}{24}}
                                                  {{1}{4}{23}}
                                                  {{2}{3}{14}}
                                                  {{2}{4}{13}}
                                                  {{3}{4}{12}}
                                                  {{1}{{2}{34}}}
                                                  {{1}{{3}{24}}}
                                                  {{1}{{4}{23}}}
                                                  {{2}{{1}{34}}}
                                                  {{2}{{3}{14}}}
                                                  {{2}{{4}{13}}}
                                                  {{3}{{1}{24}}}
                                                  {{3}{{2}{14}}}
                                                  {{3}{{4}{12}}}
                                                  {{4}{{1}{23}}}
                                                  {{4}{{2}{13}}}
                                                  {{4}{{3}{12}}}

The generalization where leaves are multisets is A330471.
The non-singleton-reduced version is A330625.
The unlabeled version is A330626.
The case with all atoms distinct is A000311.
Strongly normal multiset partitions are A035310.
Cf. A000669, A004111, A004114, A005804, A196545, A281118, A330465, A330467, A330624, A330654, A330668.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
ssrtrees[m_]:=Prepend[Join@@Table[Tuples[ssrtrees/@p],{p,Select[mps[m],Length[m]>Length[#1]>1&]}],m];
Table[Sum[Length[Select[ssrtrees[s],FreeQ[#,{_,x_Integer,x_Integer,_}]&]],{s,strnorm[n]}],{n,0,5}]

A320174 Number of series-reduced rooted trees whose leaves are constant integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 3, 6, 19, 55, 200, 713, 2740, 10651, 42637, 173012, 713280, 2972389, 12514188, 53119400, 227140464, 977382586, 4229274235, 18391269922, 80330516578, 352269725526, 1550357247476, 6845517553493, 30316222112019, 134626183784975, 599341552234773, 2674393679352974

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

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

			The a(1) = 1 through a(4) = 19 trees:
  (1)  (2)       (3)            (4)
       (11)      (111)          (22)
       ((1)(1))  ((1)(2))       (1111)
                 ((1)(11))      ((1)(3))
                 ((1)(1)(1))    ((2)(2))
                 ((1)((1)(1)))  ((2)(11))
                                ((1)(111))
                                ((11)(11))
                                ((1)(1)(2))
                                ((1)(1)(11))
                                ((1)((1)(2)))
                                ((2)((1)(1)))
                                ((1)((1)(11)))
                                ((1)(1)(1)(1))
                                ((11)((1)(1)))
                                ((1)((1)(1)(1)))
                                ((1)(1)((1)(1)))
                                (((1)(1))((1)(1)))
                                ((1)((1)((1)(1))))

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

Cf. A000081, A000311, A000669, A001678, A005804, A141268, A292504, A300660, A317099, A319312, A320173, A320175.

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]]]];
dot[m_]:=If[SameQ@@m,Prepend[#,m],#]&[Join@@Table[Union[Sort/@Tuples[dot/@p]],{p,Select[mps[m],Length[#]>1&]}]];
Table[Length[Join@@Table[dot[m],{m,IntegerPartitions[n]}]],{n,10}]

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

Terms a(11) and beyond from Andrew Howroyd, Oct 25 2018

cp's OEIS Frontend

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