A306200 -id:A306200 - cp's OEIS frontend

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

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

A306201 Number of unlabeled balanced rooted semi-identity trees with n nodes.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 6, 8, 12, 16, 25, 35, 53, 77, 117, 173, 265, 396, 605, 919, 1408, 2147, 3305, 5070, 7819, 12049, 18635, 28811, 44672, 69264, 107618, 167292, 260446, 405686, 632743, 987441, 1542555, 2411208, 3772247, 5905002, 9250436, 14499234, 22740910, 35686092

Offset: 0

Gus Wiseman, Jan 29 2019

nonn

A rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees. It is balanced if all leaves are the same distance from the root. The only balanced identity trees are rooted paths.

			The a(1) = 1 through a(7) = 8 balanced rooted semi-identity trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)      (oooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))     ((ooooo))
                 (((o)))  (((oo)))   (((ooo)))    (((oooo)))
                          ((((o))))  ((o)(oo))    ((o)(ooo))
                                     ((((oo))))   ((((ooo))))
                                     (((((o)))))  (((o)(oo)))
                                                  (((((oo)))))
                                                  ((((((o))))))

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A000081, A004111, A048816, A079500, A120803, A306200, A316624, A317712, A320169, A320270.

Row sums of A306269.

ursit[n_]:=Join@@Table[Select[Union[Sort/@Tuples[ursit/@ptn]],UnsameQ@@DeleteCases[#,{}]&],{ptn,IntegerPartitions[n-1]}];
Table[Length[Select[ursit[n],SameQ@@Length/@Position[#,{}]&]],{n,10}]

More terms from Alois P. Heinz, Jan 29 2019

A331875 Number of enriched identity p-trees of weight n.

Original entry on oeis.org

1, 1, 2, 3, 6, 14, 32, 79, 198, 522, 1368, 3716, 9992, 27612, 75692, 212045, 589478, 1668630, 4690792, 13387332, 37980664, 109098556, 311717768, 900846484, 2589449032, 7515759012, 21720369476, 63305262126, 183726039404, 537364221200, 1565570459800, 4592892152163

Offset: 1

Gus Wiseman, Jan 31 2020

nonn

An enriched identity p-tree of weight n is either the number n itself or a finite sequence of distinct enriched identity p-trees whose weights are weakly decreasing and sum to n.

			The a(1) = 1 through a(6) = 14 enriched p-trees:
  1  2  3     4        5           6
        (21)  (31)     (32)        (42)
              ((21)1)  (41)        (51)
                       ((21)2)     (321)
                       ((31)1)     ((21)3)
                       (((21)1)1)  ((31)2)
                                   ((32)1)
                                   (3(21))
                                   ((41)1)
                                   ((21)21)
                                   (((21)1)2)
                                   (((21)2)1)
                                   (((31)1)1)
                                   ((((21)1)1)1)

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

The orderless version is A300660.
The locally disjoint case is A331684.
Identity trees are A004111.
P-trees are A196545.
Enriched p-trees are A289501.
Cf. A000669, A141268, A306200, A316471, A331683, A331685, A331686, A331783.

eptrid[n_]:=Prepend[Select[Join@@Table[Tuples[eptrid/@p],{p,Rest[IntegerPartitions[n]]}],UnsameQ@@#&],n];
Table[Length[eptrid[n]],{n,10}]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(prod(k=1, n-1, sum(j=0, n\k, j!*binomial(v[k],j)*x^(k*j)) + O(x*x^n)), n)); v} \\ Andrew Howroyd, Feb 09 2020

Terms a(21) and beyond from Andrew Howroyd, Feb 09 2020

A331933 Number of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 12, 18, 33, 52, 90, 142, 242, 384, 639, 1028, 1688, 2716, 4445, 7161, 11665, 18839, 30595, 49434, 80199, 129637, 210079, 339750, 550228, 889978, 1440909, 2330887, 3772845, 6103823, 9878357, 15982196, 25863454, 41845650, 67713550, 109559443

Offset: 1

Gus Wiseman, Feb 03 2020

nonn

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

			The a(1) = 1 through a(8) = 18 trees:
  o  (o)  (oo)  (ooo)   (oooo)    (ooooo)    (oooooo)
                (o(o))  (o(oo))   (o(ooo))   (o(oooo))
                        (oo(o))   (oo(oo))   (oo(ooo))
                        ((o)(o))  (ooo(o))   (ooo(oo))
                                  (o(o)(o))  (oooo(o))
                                  (o(o(o)))  ((oo)(oo))
                                             (o(o(oo)))
                                             (o(oo(o)))
                                             (oo(o)(o))
                                             (oo(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 a(11) = 90 semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex.

Not requiring lone-child-avoidance gives A320222.
The non-semi version is A320268.
Matula-Goebel numbers of these trees are A331936.
Achiral trees are A003238.
Semi-identity trees are A306200.
Numbers S with at most one distinct prime index in S are A331912.
Semi-lone-child-avoiding rooted trees are A331934.
Cf. A000081, A001678, A004111, A050381, A214577, A291636, A320230, A320269, A331784, A331872, A331914, A331935, A331966.

sseo[n_]:=Switch[n,1,{{}},2,{{{}}},_,Join@@Function[c,Select[Union[Sort/@Tuples[sseo/@c]],Length[Union[DeleteCases[#,{}]]]<=1&]]/@Rest[IntegerPartitions[n-1]]];
Table[Length[sseo[n]],{n,10}]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sum(i=2, n-2, ((n-1)\i)*v[i])); v} \\ Andrew Howroyd, Feb 09 2020

a(n) = 1 + Sum_{i=2..n-2} floor((n-1)/i)*a(i). - Andrew Howroyd, Feb 09 2020

Terms a(31) and beyond from Andrew Howroyd, Feb 09 2020

A331783 Number of locally disjoint rooted semi-identity trees with n unlabeled vertices.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 37, 83, 191, 450, 1076, 2610, 6404, 15875, 39676, 99880, 253016, 644524, 1649918, 4242226

Offset: 1

Gus Wiseman, Jan 31 2020

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.

			The a(1) = 1 through a(6) = 17 trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (o(o))   (o(oo))    (o(ooo))
                 (((o)))  (oo(o))    (oo(oo))
                          (((oo)))   (ooo(o))
                          ((o(o)))   (((ooo)))
                          (o((o)))   ((o(oo)))
                          ((((o))))  ((oo(o)))
                                     (o((oo)))
                                     (o(o(o)))
                                     (oo((o)))
                                     ((((oo))))
                                     (((o(o))))
                                     ((o)((o)))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

The lone-child-avoiding case is A212804.
The identity tree version is A316471.
The Matula-Goebel numbers of these trees are given by A331682.
Identity trees are A004111.
Semi-identity trees are A306200.
Locally disjoint rooted trees are A316473.
Matula-Goebel numbers of locally disjoint semi-identity trees are A316494.
Cf. A000081, A306202, A316475, A316495, A316694, A331683, A331684, A331686.

disjunsQ[u_]:=Length[u]==1||UnsameQ@@DeleteCases[u,{}]&&Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
ldrsi[n_]:=If[n==1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[ldrsi/@c]]]/@IntegerPartitions[n-1],disjunsQ]];
Table[Length[ldrsi[n]],{n,10}]

A306203 Matula-Goebel numbers of balanced rooted semi-identity trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 11, 16, 17, 19, 21, 31, 32, 53, 57, 59, 64, 67, 73, 85, 127, 128, 131, 133, 159, 241, 256, 269, 277, 311, 331, 335, 365, 367, 371, 393, 399, 439, 512, 649, 709, 719, 739, 751, 917, 933, 937, 1007, 1024, 1113, 1139, 1205, 1241, 1345, 1523

Offset: 1

Gus Wiseman, Jan 29 2019

nonn

A rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees. It is balanced if all leaves are the same distance from the root. The only balanced rooted identity trees are rooted paths.

			The sequence of all unlabeled balanced rooted semi-identity trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
  11: ((((o))))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  21: ((o)(oo))
  31: (((((o)))))
  32: (ooooo)
  53: ((oooo))
  57: ((o)(ooo))
  59: ((((oo))))
  64: (oooooo)
  67: (((ooo)))
  73: (((o)(oo)))
  85: (((o))((oo)))

Cf. A000081, A004111, A007097, A048816, A184155, A276625, A306200, A306201, A306202, A316467, A317710.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
psidQ[n_]:=And[UnsameQ@@DeleteCases[primeMS[n],1],And@@psidQ/@primeMS[n]];
mgtree[n_]:=If[n==1,{},mgtree/@primeMS[n]];
Select[Range[100],And[psidQ[#],SameQ@@Length/@Position[mgtree[#],{}]]&]

A331682 One and all numbers whose prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be coprime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 22, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 40, 41, 43, 44, 47, 48, 51, 52, 53, 55, 56, 58, 59, 60, 62, 64, 66, 67, 68, 70, 71, 74, 76, 77, 79, 80, 82, 85, 86, 88, 89, 93, 94, 95, 96, 101

Offset: 1

Gus Wiseman, Jan 27 2020

nonn

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.

Also Matula-Goebel numbers of locally disjoint rooted semi-identity trees. Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex. A rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees. 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 locally disjoint rooted semi-identity trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   6: (o(o))
   7: ((oo))
   8: (ooo)
  10: (o((o)))
  11: ((((o))))
  12: (oo(o))
  13: ((o(o)))
  14: (o(oo))
  15: ((o)((o)))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  20: (oo((o)))
  22: (o(((o))))
  24: (ooo(o))

The non-semi identity tree case is A316494.
The enumeration of these trees by vertices is A331783.
Semi-identity trees are counted by A306200.
Matula-Goebel numbers of semi-identity trees are A306202.
Locally disjoint rooted trees are counted by A316473.
Matula-Goebel numbers of locally disjoint rooted trees are A316495.
Cf. A000081, A007097, A061775, A196050, A276625, A316470, A331681, A331683.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
deQ[n_]:=n==1||PrimeQ[n]&&deQ[PrimePi[n]]||CoprimeQ@@primeMS[n]&&And@@deQ/@primeMS[n];
Select[Range[100],deQ]

A331994 Matula-Goebel numbers of semi-lone-child-avoiding rooted semi-identity trees.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 16, 21, 24, 26, 28, 32, 38, 39, 42, 48, 52, 56, 57, 64, 74, 76, 78, 84, 86, 91, 96, 104, 106, 111, 112, 114, 128, 129, 133, 146, 148, 152, 156, 159, 168, 172, 178, 182, 192, 202, 208, 212, 214, 219, 222, 224, 228, 247, 256, 258, 259, 262

Offset: 1

Gus Wiseman, Feb 05 2020

nonn

Semi-lone-child-avoiding means there are no vertices with exactly one child unless that child is an endpoint/leaf.
In a semi-identity tree, the non-leaf branches of any given vertex are distinct.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
Consists of one, two, and all numbers that can be written as a power of two (other than 2) times a squarefree number whose prime indices already belong to the sequence, where a prime index of n is a number m such that prime(m) divides n.

			The sequence of all semi-lone-child-avoiding rooted semi-identity trees 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)
  21: ((o)(oo))
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  32: (ooooo)
  38: (o(ooo))
  39: ((o)(o(o)))
  42: (o(o)(oo))
  48: (oooo(o))
  52: (oo(o(o)))
  56: (ooo(oo))
  57: ((o)(ooo))
The sequence of terms together with their prime indices begins:
    1: {}              64: {1,1,1,1,1,1}      159: {2,16}
    2: {1}             74: {1,12}             168: {1,1,1,2,4}
    4: {1,1}           76: {1,1,8}            172: {1,1,14}
    6: {1,2}           78: {1,2,6}            178: {1,24}
    8: {1,1,1}         84: {1,1,2,4}          182: {1,4,6}
   12: {1,1,2}         86: {1,14}             192: {1,1,1,1,1,1,2}
   14: {1,4}           91: {4,6}              202: {1,26}
   16: {1,1,1,1}       96: {1,1,1,1,1,2}      208: {1,1,1,1,6}
   21: {2,4}          104: {1,1,1,6}          212: {1,1,16}
   24: {1,1,1,2}      106: {1,16}             214: {1,28}
   26: {1,6}          111: {2,12}             219: {2,21}
   28: {1,1,4}        112: {1,1,1,1,4}        222: {1,2,12}
   32: {1,1,1,1,1}    114: {1,2,8}            224: {1,1,1,1,1,4}
   38: {1,8}          128: {1,1,1,1,1,1,1}    228: {1,1,2,8}
   39: {2,6}          129: {2,14}             247: {6,8}
   42: {1,2,4}        133: {4,8}              256: {1,1,1,1,1,1,1,1}
   48: {1,1,1,1,2}    146: {1,21}             258: {1,2,14}
   52: {1,1,6}        148: {1,1,12}           259: {4,12}
   56: {1,1,1,4}      152: {1,1,1,8}          262: {1,32}
   57: {2,8}          156: {1,1,2,6}          266: {1,4,8}

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

The locally disjoint version is A331681.
The enumeration of these trees by vertices is A331993.
Semi-identity trees are A306200.
MG-numbers of rooted identity trees are A276625.
MG-numbers of lone-child-avoiding rooted identity trees are {1}.
MG-numbers of lone-child-avoiding rooted trees are A291636.
MG-numbers of semi-identity trees are A306202.
MG-numbers of semi-lone-child-avoiding rooted trees are A331935.
MG-numbers of semi-lone-child-avoiding rooted identity trees are A331963.
MG-numbers of lone-child-avoiding rooted semi-identity trees are A331965.
Cf. A004111, A007097, A061775, A122132, A196050, A320269, A331683, A331873, A331934, A331936, A331964, A331966.

scsiQ[n_]:=n==1||n==2||!PrimeQ[n]&&FreeQ[FactorInteger[n],{?(#>2&),?(#>1&)}]&&And@@scsiQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],scsiQ]

Intersection of A306202 and A331935.

A331993 Number of semi-lone-child-avoiding rooted semi-identity trees with n unlabeled vertices.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 11, 22, 43, 90, 185, 393, 835, 1802, 3904, 8540, 18756, 41463, 92022, 205179, 459086, 1030917, 2321949, 5245104, 11878750, 26967957, 61359917, 139902251, 319591669, 731385621, 1676573854, 3849288924, 8850674950, 20378544752, 46982414535

Offset: 1

Gus Wiseman, Feb 05 2020

nonn

Semi-lone-child-avoiding means there are no vertices with exactly one child unless that child is an endpoint/leaf.

In a semi-identity tree, the non-leaf branches of any given vertex are distinct.

			The a(1) = 1 through a(7) = 11 trees:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)    (oooooo)
                (o(o))  (o(oo))  (o(ooo))   (o(oooo))
                        (oo(o))  (oo(oo))   (oo(ooo))
                                 (ooo(o))   (ooo(oo))
                                 ((o)(oo))  (oooo(o))
                                 (o(o(o)))  ((o)(ooo))
                                            (o(o)(oo))
                                            (o(o(oo)))
                                            (o(oo(o)))
                                            (oo(o(o)))
                                            ((o)(o(o)))

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Not requiring any lone-child-avoidance gives A306200.
The locally disjoint case is A324969 (essentially A000045).
Matula-Goebel numbers of these trees are A331994.
Lone-child-avoiding rooted identity trees are A000007.
Semi-lone-child-avoiding rooted trees are A331934.
Semi-lone-child-avoiding rooted identity trees are A331964.
Lone-child-avoiding rooted semi-identity trees are A331966.
Cf. A001678, A004111, A300660, A316694, A331683, A331686, A331783, A331875, A331933, A331963, A331965.

sssb[n_]:=Switch[n,1,{{}},2,{{{}}},_,Join@@Function[c,Select[Union[Sort/@Tuples[sssb/@c]],UnsameQ@@DeleteCases[#,{}]&]]/@Rest[IntegerPartitions[n-1]]];
Table[Length[sssb[n]],{n,10}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=[0]); for(n=1, n-1, v=concat(v, 1 + vecsum(WeighT(v)) - v[n])); v[1]=1; v} \\ Andrew Howroyd, Feb 09 2020

Terms a(26) and beyond from Andrew Howroyd, Feb 09 2020

A339193 Matula-Goebel numbers of unlabeled binary rooted semi-identity trees.

Original entry on oeis.org

1, 4, 14, 86, 301, 886, 3101, 3986, 13766, 13951, 19049, 48181, 57026, 75266, 85699, 199591, 263431, 295969, 298154, 302426, 426058, 882899

Offset: 1

Gus Wiseman, Mar 14 2021

nonn

Definition: A positive integer belongs to the sequence iff it is 1, 4, or a squarefree semiprime whose prime indices both already belong to the sequence. 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.
In a semi-identity tree, only the non-leaf branches of any given vertex are distinct. Alternatively, a rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees.
The Matula-Goebel number of an unlabeled rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The sequence of terms together with the corresponding unlabeled rooted trees begins:
      1: o
      4: (oo)
     14: (o(oo))
     86: (o(o(oo)))
    301: ((oo)(o(oo)))
    886: (o(o(o(oo))))
   3101: ((oo)(o(o(oo))))
   3986: (o((oo)(o(oo))))
  13766: (o(o(o(o(oo)))))
  13951: ((oo)((oo)(o(oo))))
  19049: ((o(oo))(o(o(oo))))
  48181: ((oo)(o(o(o(oo)))))
  57026: (o((oo)(o(o(oo)))))
  75266: (o(o((oo)(o(oo)))))
  85699: ((o(oo))((oo)(o(oo))))

Gus Wiseman, The sequence of all unlabeled binary rooted semi-identity trees by Matula-Goebel number.

Counting these trees by number of nodes gives A063895.
A000081 counts unlabeled rooted trees with n nodes.
A111299 ranks binary trees, counted by A001190.
A276625 ranks identity trees, counted by A004111.
A306202 ranks semi-identity trees, counted by A306200.
A306203 ranks balanced semi-identity trees, counted by A306201.
A331965 ranks lone-child avoiding semi-identity trees, counted by A331966.
Cf. A007097, A061775, A196050, A291636, A331963, A331964.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mgbiQ[n_]:=Or[n==1,n==4,SquareFreeQ[n]&&PrimeOmega[n]==2&&And@@mgbiQ/@primeMS[n]];
Select[Range[1000],mgbiQ]

cp's OEIS Frontend

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

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A306201 Number of unlabeled balanced rooted semi-identity trees with n nodes.

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A331875 Number of enriched identity p-trees of weight n.

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A331933 Number of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex.

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A331783 Number of locally disjoint rooted semi-identity trees with n unlabeled vertices.

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A306203 Matula-Goebel numbers of balanced rooted semi-identity trees.

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A331682 One and all numbers whose prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be coprime.

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A331994 Matula-Goebel numbers of semi-lone-child-avoiding rooted semi-identity trees.

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