A048816 -id:A048816 - cp's OEIS frontend

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

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.

A320270 Number of unlabeled balanced semi-binary rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 7, 10, 13, 19, 25, 35, 46, 65, 88, 124, 171, 242, 334, 470, 653, 921, 1287, 1822, 2565, 3640, 5144, 7311, 10360, 14734, 20918, 29781, 42361, 60389, 86069, 122893, 175479, 250922, 358863

Offset: 1

Gus Wiseman, Oct 08 2018

nonn

An unlabeled rooted tree is semi-binary if all out-degrees are <= 2, and balanced if all leaves are the same distance from the root. The number of semi-binary trees with n nodes is equal to the number of binary trees with n+1 leaves; see A001190.

			The a(1) = 1 through a(7) = 6 balanced semi-binary rooted trees:
  o  (o)  (oo)   ((oo))   (((oo)))   ((o)(oo))    ((oo)(oo))
          ((o))  (((o)))  ((o)(o))   ((((oo))))   (((o)(oo)))
                          ((((o))))  (((o)(o)))   (((((oo)))))
                                     (((((o)))))  ((((o)(o))))
                                                  (((o))((o)))
                                                  ((((((o))))))

Gus Wiseman, The a(13) = 35 balanced semi-binary rooted trees.
Gus Wiseman, The a(15) = 65 balanced semi-binary rooted trees.
Gus Wiseman, The a(16) = 88 balanced semi-binary rooted trees.
Gus Wiseman, The a(18) = 171 balanced semi-binary rooted trees.

Cf. A001190, A048816, A079500, A111299, A292050, A298204, A301345, A320271.

saur[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[saur/@ptn]],SameQ@@Length/@Position[#,{}]&],{ptn,Select[IntegerPartitions[n-1],Length[#]<=2&]}]];
Table[Length[saur[n]],{n,20}]

A330666 Number of non-isomorphic balanced reduced multisystems whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 2, 10, 11, 20, 15, 90, 51, 80, 6, 468, 93, 2910, 80, 521, 277, 20644, 80, 334, 1761, 393, 521, 165874, 1374

Offset: 1

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 prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			Non-isomorphic representatives of the a(2) = 1 through a(9) = 10 multisystems (commas and outer brackets elided):
    1  11  12  111      112      1111            123      1122
               {1}{11}  {1}{12}  {1}{111}        {1}{23}  {1}{122}
                        {2}{11}  {11}{11}                 {11}{22}
                                 {1}{1}{11}               {12}{12}
                                 {{1}}{{1}{11}}           {1}{1}{22}
                                 {{11}}{{1}{1}}           {1}{2}{12}
                                                          {{1}}{{1}{22}}
                                                          {{11}}{{2}{2}}
                                                          {{1}}{{2}{12}}
                                                          {{12}}{{1}{2}}
Non-isomorphic representatives of the a(12) = 15 multisystems:
  {1,1,2,3}
  {{1},{1,2,3}}
  {{1,1},{2,3}}
  {{1,2},{1,3}}
  {{2},{1,1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{2},{3},{1,1}}
  {{{1}},{{1},{2,3}}}
  {{{1,1}},{{2},{3}}}
  {{{1}},{{2},{1,3}}}
  {{{1,2}},{{1},{3}}}
  {{{2}},{{1},{1,3}}}
  {{{2}},{{3},{1,1}}}
  {{{2,3}},{{1},{1}}}

The labeled version is A318846.
The maximum-depth version is A330664.
Unlabeled balanced reduced multisystems by weight are A330474.
The case of constant or strict atoms is A318813.
Cf. A000669, A005121, A007716, A048816, A141268, A306186, A317791, A318812, A318849, A330470, A330475, A330655, A330728.

a(2^n) = a(prime(n)) = A318813(n).

A048809 Number of rooted trees with n nodes with every leaf at height 4.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 15, 23, 39, 61, 102, 161, 265, 420, 682, 1087, 1753, 2790, 4476, 7120, 11376, 18075, 28785, 45666, 72530, 114882, 182040, 287878, 455231, 718755, 1134491, 1788461, 2818140, 4436000, 6978932, 10969695, 17232572, 27049320

Offset: 5

Christian G. Bower, Apr 15 1999

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A048808-A048816.

Column k=4 of A244925.

Euler transform of A048808 shifted right.

A048815 Number of rooted trees with n nodes with every leaf at height 10.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 28, 49, 83, 145, 245, 425, 724, 1246, 2130, 3659, 6254, 10724, 18335, 31396, 53676, 91832, 156944, 268324, 458435, 783324, 1337862, 2284950, 3901211, 6660304, 11367935, 19401130, 33104598, 56481086, 96349147

Offset: 11

Christian G. Bower, Apr 15 1999

nonn

Alois P. Heinz, Table of n, a(n) for n = 11..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A048808-A048816.

Column k=10 of A244925.

Euler transform of A048814 shifted right.

A320172 Number of series-reduced balanced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 2, 5, 9, 19, 38, 79, 163, 352, 750, 1633, 3558, 7783, 17020, 37338, 81920, 180399, 398600, 885101, 1975638, 4435741, 10013855, 22726109, 51807432, 118545425, 272024659, 625488420, 1440067761, 3317675261, 7644488052, 17610215982, 40547552277, 93298838972, 214516498359, 492844378878

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. In an identity tree, all branches directly under any given node are different.

			The a(1) = 1 through a(5) = 19 rooted identity trees:
  (1)  (2)   (3)        (4)         (5)
       (11)  (21)       (22)        (32)
             (111)      (31)        (41)
             ((1)(2))   (211)       (221)
             ((1)(11))  (1111)      (311)
                        ((1)(3))    (2111)
                        ((1)(21))   (11111)
                        ((2)(11))   ((1)(4))
                        ((1)(111))  ((2)(3))
                                    ((1)(31))
                                    ((1)(22))
                                    ((2)(21))
                                    ((3)(11))
                                    ((1)(211))
                                    ((11)(21))
                                    ((2)(111))
                                    ((1)(1111))
                                    ((11)(111))
                                    ((1)(2)(11))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000669, A005804, A048816, A079500, A119262, A120803, A141268, A292504, A300660, A319312.

Cf. A320154, A320155, A320160, A320171, A320173, A320177, A320178, 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]]]];
gig[m_]:=Prepend[Join@@Table[Union[Sort/@Select[Sort/@Tuples[gig/@mtn],UnsameQ@@#&]],{mtn,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[Select[gig[y],SameQ@@Length/@Position[#,_Integer]&]],{y,Sort /@IntegerPartitions[n]}],{n,8}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(u=vector(n, n, numbpart(n)), v=vector(n)); while(u, v+=u; u=WeighT(u)-u); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(13) and beyond from Andrew Howroyd, Oct 25 2018

A330627 Number of non-isomorphic phylogenetic trees with n nodes.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 4, 5, 9, 14, 24, 39, 69, 116, 205, 357, 632, 1118, 2001, 3576, 6445, 11627, 21080, 38293, 69819, 127539, 233644, 428825, 788832, 1453589, 2683602, 4962167, 9190155, 17044522, 31655676, 58866237, 109600849, 204293047, 381212823, 712073862

Offset: 1

Gus Wiseman, Dec 28 2019

nonn

A phylogenetic tree is a series-reduced rooted tree whose leaves are (usually disjoint) sets. Each branching as well as each element of each leaf contributes to the number of nodes.

			Non-isomorphic representatives of the a(2) = 1 through a(9) = 9 trees (commas and outer brackets elided):
  1  12  123  1234    12345    123456     1234567      12345678
              (1)(2)  (1)(23)  (1)(234)   (1)(2345)    (1)(23456)
                               (12)(34)   (12)(345)    (12)(3456)
                               (1)(2)(3)  (1)(2)(34)   (123)(456)
                                          (1)((2)(3))  (1)(2)(345)
                                                       (1)(23)(45)
                                                       (1)((2)(34))
                                                       (1)(2)(3)(4)
                                                       (12)((3)(4))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Phylogenetic trees by number of labels are A005804, with unlabeled version A141268.
Balanced phylogenetic trees are A320154.
Cf. A000311, A000669, A001678, A004114, A005121, A007716, A048816, A060356, A330465, A330467, A330469.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=[0]); for(n=1, n-1, v=concat(v, EulerT(v)[n] - v[n] + 1)); v} \\ Andrew Howroyd, Jan 02 2021

G.f.: A(x) satisfies A(x) = x*(1/(1-x) - A(x) - 2 + exp(Sum_{k>0} A(x^k)/k)). - Andrew Howroyd, Jan 02 2021

Terms a(11) and beyond from Andrew Howroyd, Jan 02 2021

A048810 Number of rooted trees with n nodes with every leaf at height 5.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 16, 26, 44, 73, 123, 203, 340, 563, 935, 1550, 2571, 4251, 7034, 11618, 19188, 31654, 52201, 85999, 141631, 233074, 383375, 630215, 1035508, 1700501, 2791309, 4579587, 7510280, 12310980, 20172075, 33039130, 54092556

Offset: 6

Christian G. Bower, Apr 15 1999

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A048808-A048816.

Column k=5 of A244925.

Euler transform of A048809 shifted right.

A048811 Number of rooted trees with n nodes with every leaf at height 6.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 27, 47, 78, 135, 224, 384, 642, 1088, 1827, 3088, 5182, 8736, 14661, 24660, 41378, 69500, 116534, 195509, 327627, 549104, 919593, 1539985, 2577399, 4313102, 7214374, 12064930, 20169283, 33710370, 56324729, 94089240

Offset: 7

Christian G. Bower, Apr 15 1999

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A048808-A048816.

Column k=6 of A244925.

Euler transform of A048810 shifted right.

A048812 Number of rooted trees with n nodes with every leaf at height 7.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 28, 48, 81, 140, 236, 405, 686, 1169, 1984, 3375, 5723, 9721, 16478, 27949, 47354, 80245, 135869, 230054, 389304, 658706, 1114072, 1883900, 3184602, 5382321, 9094154, 15362767, 25946131, 43811971, 73964065

Offset: 8

Christian G. Bower, Apr 15 1999

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A048808-A048816.

Column k=7 of A244925.

Euler transform of A048811 shifted right.

cp's OEIS Frontend

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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A320270 Number of unlabeled balanced semi-binary rooted trees with n nodes.

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A330666 Number of non-isomorphic balanced reduced multisystems whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.

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A048809 Number of rooted trees with n nodes with every leaf at height 4.

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A048815 Number of rooted trees with n nodes with every leaf at height 10.

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A320172 Number of series-reduced balanced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

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A330627 Number of non-isomorphic phylogenetic trees with n nodes.

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A048810 Number of rooted trees with n nodes with every leaf at height 5.

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A048811 Number of rooted trees with n nodes with every leaf at height 6.

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A048812 Number of rooted trees with n nodes with every leaf at height 7.

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