A330624 -id:A330624 - cp's OEIS frontend

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

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

Offset: 0

Vladeta Jovovic, Apr 01 2001

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

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

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

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

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

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

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

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

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

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

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

Original entry on oeis.org

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

Offset: 1

Vladeta Jovovic, Jul 20 2005

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

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

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

The rooted version is A060356.

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

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

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

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

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

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 21 2019

nonn

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

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

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

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

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

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

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}]

A330668 Number of non-isomorphic balanced reduced multisystems of weight n whose leaves (which are multisets of atoms) are all sets.

Original entry on oeis.org

1, 1, 1, 3, 22, 204, 2953

Offset: 0

Gus Wiseman, Dec 27 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. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 22 multisystems:
  {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},{1}}}
                           {{{1}},{{2},{1,2}}}
                           {{{1,2}},{{1},{2}}}
                           {{{1}},{{2},{1,3}}}
                           {{{1,2}},{{1},{3}}}
                           {{{1}},{{2},{3,4}}}
                           {{{1,2}},{{3},{4}}}
                           {{{2}},{{1},{1,3}}}
                           {{{2,3}},{{1},{1}}}

The case with all atoms different is A318813.
The version where the leaves are multisets is A330474.
The tree version is A330626.
The maximum-depth case is A330677.
Unlabeled series-reduced rooted trees whose leaves are sets are A330624.
Cf. A000311, A004114, A005121, A005804, A007716, A048816, A141268, A283877, A306186, A318812, A320154, A330470, A330628, A330663.

A330677 Number of non-isomorphic balanced reduced multisystems of weight n and maximum depth whose leaves (which are multisets of atoms) are sets.

Original entry on oeis.org

1, 1, 1, 2, 11, 81, 859

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. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

			Non-isomorphic representatives of the a(0) = 1 through a(4) = 11 multisystems:
  {}  {1}  {1,2}  {{1},{1,2}}  {{{1}},{{1},{1,2}}}
                  {{1},{2,3}}  {{{1}},{{1},{2,3}}}
                               {{{1,2}},{{1},{1}}}
                               {{{1}},{{2},{1,2}}}
                               {{{1,2}},{{1},{2}}}
                               {{{1}},{{2},{1,3}}}
                               {{{1,2}},{{1},{3}}}
                               {{{1}},{{2},{3,4}}}
                               {{{1,2}},{{3},{4}}}
                               {{{2}},{{1},{1,3}}}
                               {{{2,3}},{{1},{1}}}

The version with all distinct atoms is A000111.
Non-isomorphic set multipartitions are A049311.
The (non-maximal) tree version is A330626.
Allowing leaves to be multisets gives A330663.
The case with prescribed degrees is A330664.
The version allowing all depths is A330668.
Cf. A000669, A001678, A004114, A005121, A007716, A141268, A283877, A306186, A330465, A330470, A330624.

cp's OEIS Frontend

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

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

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

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

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

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

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A330668 Number of non-isomorphic balanced reduced multisystems of weight n whose leaves (which are multisets of atoms) are all sets.

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