A331935 -id:A331935 - cp's OEIS frontend

A111299 Numbers whose Matula tree is a binary tree (i.e., root has degree 2 and all nodes except root and leaves have degree 3).

4, 14, 49, 86, 301, 454, 886, 1589, 1849, 3101, 3986, 6418, 9761, 13766, 13951, 19049, 22463, 26798, 31754, 48181, 51529, 57026, 75266, 85699, 93793, 100561, 111139, 128074, 137987, 196249, 199591, 203878, 263431, 295969, 298154, 302426, 426058, 448259, 452411

Offset: 1

Keith Briggs, Nov 02 2005

nonn

This sequence should probably start with 1. Then a number k is in the sequence iff k = 1 or k = prime(x) * prime(y) with x and y already in the sequence. - Gus Wiseman, May 04 2021

			From _Gus Wiseman_, May 04 2021: (Start)
The sequence of trees (starting with 1) begins:
     1: o
     4: (oo)
    14: (o(oo))
    49: ((oo)(oo))
    86: (o(o(oo)))
   301: ((oo)(o(oo)))
   454: (o((oo)(oo)))
   886: (o(o(o(oo))))
  1589: ((oo)((oo)(oo)))
  1849: ((o(oo))(o(oo)))
  3101: ((oo)(o(o(oo))))
  3986: (o((oo)(o(oo))))
  6418: (o(o((oo)(oo))))
  9761: ((o(oo))((oo)(oo)))
(End)

Kevin Ryde, Table of n, a(n) for n = 1..5000 (terms 1..186 from Charles R Greathouse IV)
Keith Briggs, Matula numbers and rooted trees
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Kevin Ryde, PARI/GP Code
Index entries for sequences related to Matula-Goebel numbers

Cf. A245824 (by number of leaves).
Cf. A005517, A005518, A061773, A245824.
These trees are counted by 2*A001190 - 1.
The semi-binary version is A292050 (counted by A001190).
The semi-identity case is A339193 (counted by A063895).
A000081 counts unlabeled rooted trees with n nodes.
A007097 ranks rooted chains.
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. A036656, A061775, A196050, A291636, A320230, A331935, A331965.

nn=20000;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
binQ[n_]:=Or[n===1,With[{m=primeMS[n]},And[Length[m]===2,And@@binQ/@m]]];
Select[Range[2,nn],binQ] (* Gus Wiseman, Aug 28 2017 *)

i(n)=n==2 || is(primepi(n))
is(n)=if(n<14,return(n==4)); my(f=factor(n),t=#f[,1]); if(t>1, t==2 && f[1,2]==1 && f[2,2]==1 && i(f[1,1]) && i(f[2,1]), f[1,2]==2 && i(f[1,1])) \\ Charles R Greathouse IV, Mar 29 2013

list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, if(i(p)&&i(q), listput(v, t*q)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Mar 29 2013

PARI
```
\\ Also see links.
```

The Matula tree of k is defined as follows:
matula(k):
    create a node labeled k
    for each prime factor m of k:
       add the subtree matula(prime(m)), by an edge labeled m
    return the node

Definition corrected by Charles R Greathouse IV, Mar 29 2013

a(27)-a(39) from Charles R Greathouse IV, Mar 29 2013

A050381 Number of series-reduced planted trees with n leaves of 2 colors.

Original entry on oeis.org

2, 3, 10, 40, 170, 785, 3770, 18805, 96180, 502381, 2667034, 14351775, 78096654, 429025553, 2376075922, 13252492311, 74372374366, 419651663108, 2379399524742, 13549601275893, 77460249369658, 444389519874841

Offset: 1

Christian G. Bower, Nov 15 1999

nonn

Consider the free algebraic system with two commutative associative operators (x+y) and (x*y) and two generators A,B. The number of elements with n occurrences of the generators is 2*a(n) if n>1, and the number of generators if n=1. - Michael Somos, Aug 07 2017
From Gus Wiseman, Feb 07 2020: (Start)
Also the number of semi-lone-child-avoiding rooted trees with n leaves. Semi-lone-child-avoiding means there are no vertices with exactly one child unless that child is an endpoint/leaf. For example, the a(1) = 2 through a(3) = 10 trees are:
  o    (oo)      (ooo)
  (o)  (o(o))    (o(oo))
       ((o)(o))  (oo(o))
                 ((o)(oo))
                 (o(o)(o))
                 (o(o(o)))
                 ((o)(o)(o))
                 ((o)(o(o)))
                 (o((o)(o)))
                 ((o)((o)(o)))
(End)

			For n=2, the 2*a(2) = 6 elements are: A+A, A+B, B+B, A*A, A*B, B*B. - _Michael Somos_, Aug 07 2017

Andrew Howroyd, Table of n, a(n) for n = 1..500
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012. - From _N. J. A. Sloane_, Dec 22 2012
N. J. A. Sloane, Transforms
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
Index entries for sequences related to rooted trees

Column 2 of A319254.
Cf. A029856, A031148.
Lone-child-avoiding rooted trees with n leaves are A000669.
Lone-child-avoiding rooted trees with n vertices are A001678.
The locally disjoint case is A331874.
Semi-lone-child-avoiding rooted trees with n vertices are A331934.
Matula-Goebel numbers of these trees are A331935.
Cf. A000081, A005804, A141268, A196545, A291636, A316697, A330465, A331872, A331933, A331964.

terms = 22;
B[x_] = x O[x]^(terms+1);
A[x_] = 1/(1 - x + B[x])^2;
Do[A[x_] = A[x]/(1 - x^k + B[x])^Coefficient[A[x], x, k] + O[x]^(terms+1) // Normal, {k, 2, terms+1}];
Join[{2}, Drop[CoefficientList[A[x], x]/2, 2]] (* Jean-François Alcover, Aug 17 2018, after Michael Somos *)
slaurte[n_]:=If[n==1,{o,{o}},Join@@Table[Union[Sort/@Tuples[slaurte/@ptn]],{ptn,Rest[IntegerPartitions[n]]}]];
Table[Length[slaurte[n]],{n,10}] (* Gus Wiseman, Feb 07 2020 *)

{a(n) = my(A, B); if( n<2, 2*(n>0), B = x * O(x^n); A = 1 / (1 - x + B)^2; for(k=2, n, A /= (1 - x^k + B)^polcoeff(A, k)); polcoeff(A, n)/2)}; /* Michael Somos, Aug 07 2017 */

Doubles (index 2+) under EULER transform.
Product_{k>=1} (1-x^k)^-a(k) = 1 + a(1)*x + Sum_{k>=2} 2*a(k)*x^k. - Michael Somos, Aug 07 2017
a(n) ~ c * d^n / n^(3/2), where d = 6.158893517087396289837838459951206775682824030495453326610366016992093939... and c = 0.1914250508201011360729769525164141605187995730026600722369002... - Vaclav Kotesovec, Aug 17 2018

A331934 Number of semi-lone-child-avoiding rooted trees with n unlabeled vertices.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 15, 29, 62, 129, 279, 602, 1326, 2928, 6544, 14692, 33233, 75512, 172506, 395633, 911108, 2105261, 4880535, 11346694, 26451357, 61813588, 144781303, 339820852, 799168292, 1882845298, 4443543279, 10503486112, 24864797324, 58944602767, 139918663784

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(7) = 15 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)(oo))  (oooo(o))
                                  (o(o)(o))  ((o)(ooo))
                                  (o(o(o)))  ((oo)(oo))
                                             (o(o)(oo))
                                             (o(o(oo)))
                                             (o(oo(o)))
                                             (oo(o)(o))
                                             (oo(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.

The same trees counted by leaves are A050381.
The locally disjoint version is A331872.
Matula-Goebel numbers of these trees are A331935.
Lone-child-avoiding rooted trees are A001678.
Cf. A000081, A000669, A198518, A289501, A291636, A306200, A320268, A330465, A330951, A331873, A331874, A331933, A331966.

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

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

Product_{k > 0} 1/(1 - x^k)^a(k) = A(x) + A(x)/x - x where A(x) = Sum_{k > 0} x^k a(k).

Euler transform is b(1) = 1, b(n > 1) = a(n) + a(n + 1).

Terms a(25) and beyond from Andrew Howroyd, Feb 09 2020

A331873 Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted trees.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 14, 16, 18, 24, 26, 27, 28, 32, 36, 38, 46, 48, 49, 52, 54, 56, 64, 69, 72, 74, 76, 81, 86, 92, 96, 98, 104, 106, 108, 112, 122, 128, 138, 144, 148, 152, 161, 162, 169, 172, 178, 184, 192, 196, 202, 206, 207, 208, 212, 214, 216, 224, 243

Offset: 1

Gus Wiseman, Feb 02 2020

nonn

First differs from A331936 in having 69, the Matula-Goebel number of the tree ((o)((o)(o))).
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 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 nonprime numbers whose distinct prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be pairwise coprime. A prime index of n is a number m such that prime(m) divides n.

			The sequence of all semi-lone-child-avoiding locally disjoint rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
   9: ((o)(o))
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  18: (o(o)(o))
  24: (ooo(o))
  26: (o(o(o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  46: (o((o)(o)))
  48: (oooo(o))
  49: ((oo)(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.

Not requiring lone-child-avoidance gives A316495.
A superset of A320269.
The semi-identity tree case is A331681.
The non-semi version (i.e., not containing 2) is A331871.
These trees counted by vertices are A331872.
These trees counted by leaves are A331874.
Not requiring local disjointness gives A331935.
The identity tree case is A331937.
Cf. A007097, A050381, A061775, A196050, A291636, A302696, A316473, A316696, A316697, A331680, A331682, A331683, A331687, A331934.

msQ[n_]:=n==1||n==2||!PrimeQ[n]&&(PrimePowerQ[n]||CoprimeQ@@PrimePi/@First/@FactorInteger[n])&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],msQ]

A331936 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex (semi-achirality).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 14, 16, 18, 24, 26, 27, 28, 32, 36, 38, 46, 48, 49, 52, 54, 56, 64, 72, 74, 76, 81, 86, 92, 96, 98, 104, 106, 108, 112, 122, 128, 144, 148, 152, 162, 169, 172, 178, 184, 192, 196, 202, 206, 208, 212, 214, 216, 224, 243, 244, 256, 262, 288

Offset: 1

Gus Wiseman, Feb 03 2020

nonn

First differs from A331873 in lacking 69, the Matula-Goebel number of the tree ((o)((o)(o))).
A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf.
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 1, 2, and all numbers equal to a power of 2 (other than 1) times a power of prime(j) for some j > 1 already in the sequence.

			The sequence of rooted trees ranked by this sequence together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
   9: ((o)(o))
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  18: (o(o)(o))
  24: (ooo(o))
  26: (o(o(o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  46: (o((o)(o)))
  48: (oooo(o))
  49: ((oo)(oo))
The sequence of terms together with their prime indices begins:
    1: {}              52: {1,1,6}            152: {1,1,1,8}
    2: {1}             54: {1,2,2,2}          162: {1,2,2,2,2}
    4: {1,1}           56: {1,1,1,4}          169: {6,6}
    6: {1,2}           64: {1,1,1,1,1,1}      172: {1,1,14}
    8: {1,1,1}         72: {1,1,1,2,2}        178: {1,24}
    9: {2,2}           74: {1,12}             184: {1,1,1,9}
   12: {1,1,2}         76: {1,1,8}            192: {1,1,1,1,1,1,2}
   14: {1,4}           81: {2,2,2,2}          196: {1,1,4,4}
   16: {1,1,1,1}       86: {1,14}             202: {1,26}
   18: {1,2,2}         92: {1,1,9}            206: {1,27}
   24: {1,1,1,2}       96: {1,1,1,1,1,2}      208: {1,1,1,1,6}
   26: {1,6}           98: {1,4,4}            212: {1,1,16}
   27: {2,2,2}        104: {1,1,1,6}          214: {1,28}
   28: {1,1,4}        106: {1,16}             216: {1,1,1,2,2,2}
   32: {1,1,1,1,1}    108: {1,1,2,2,2}        224: {1,1,1,1,1,4}
   36: {1,1,2,2}      112: {1,1,1,1,4}        243: {2,2,2,2,2}
   38: {1,8}          122: {1,18}             244: {1,1,18}
   46: {1,9}          128: {1,1,1,1,1,1,1}    256: {1,1,1,1,1,1,1,1}
   48: {1,1,1,1,2}    144: {1,1,1,1,2,2}      262: {1,32}
   49: {4,4}          148: {1,1,12}           288: {1,1,1,1,1,2,2}

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.

A superset of A000079.
The non-lone-child-avoiding version is A320230.
The non-semi version is A320269.
These trees are counted by A331933.
Not requiring semi-achirality gives A331935.
The fully-achiral case is A331992.
Achiral trees are counted by A003238.
Numbers with at most one distinct odd prime factor are A070776.
Matula-Goebel numbers of achiral rooted trees are A214577.
Matula-Goebel numbers of semi-identity trees are A306202.
Numbers S with at most one distinct prime index in S are A331912.
Cf. A001678, A007097, A050381, A061775, A196050, A291636, A331784, A331873, A331914, A331934, A331965, A331967, A331991.

msQ[n_]:=n<=2||!PrimeQ[n]&&Length[DeleteCases[FactorInteger[n],{2,_}]]<=1&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],msQ]

Intersection of A320230 and A331935.

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

A331963 Matula-Goebel numbers of semi-lone-child-avoiding rooted identity trees.

Original entry on oeis.org

1, 2, 6, 26, 39, 78, 202, 303, 334, 501, 606, 794, 1002, 1191, 1313, 2171, 2382, 2462, 2626, 3693, 3939, 3998, 4342, 4486, 5161, 5997, 6513, 6729, 7162, 7386, 7878, 8914, 10322, 10743, 11994, 12178, 13026, 13371, 13458, 15483, 15866, 16003, 16867, 18267, 19286

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. It is an identity tree if the branches under any given vertex are all 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 nonprime squarefree numbers 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 identity trees together with their Matula-Goebel numbers begins:
    1: o
    2: (o)
    6: (o(o))
   26: (o(o(o)))
   39: ((o)(o(o)))
   78: (o(o)(o(o)))
  202: (o(o(o(o))))
  303: ((o)(o(o(o))))
  334: (o((o)(o(o))))
  501: ((o)((o)(o(o))))
  606: (o(o)(o(o(o))))
  794: (o(o(o)(o(o))))

Gus Wiseman, The first 63 semi-lone-child-avoiding rooted identity trees.

A subset of A276625 (MG-numbers of identity trees).
Not requiring an identity tree gives A331935.
The locally disjoint version is A331937.
These trees are counted by A331964.
The semi-identity case is A331994.
Matula-Goebel numbers of identity trees are A276625.
Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees are A331965.
Cf. A004111, A007097, A050381, A061775, A196050, A291636, A300660, A306202, A320269, A331681, A331873, A331875, A331933, A331934, A331936.

msiQ[n_]:=n==1||n==2||!PrimeQ[n]&&SquareFreeQ[n]&&And@@msiQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[1000],msiQ]

Intersection of A276625 (identity trees) and A331935 (semi-lone-child-avoiding).

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

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}]

A331874 Number of semi-lone-child-avoiding locally disjoint rooted trees with n unlabeled leaves.

Original entry on oeis.org

2, 3, 8, 24, 67, 214, 687, 2406, 8672, 32641, 125431, 493039, 1964611

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) = 2 through a(4) = 24 trees:
  o    (oo)      (ooo)          (oooo)
  (o)  (o(o))    (o(oo))        (o(ooo))
       ((o)(o))  (oo(o))        (oo(oo))
                 (o(o)(o))      (ooo(o))
                 (o(o(o)))      ((oo)(oo))
                 ((o)(o)(o))    (o(o(oo)))
                 (o((o)(o)))    (o(oo(o)))
                 ((o)((o)(o)))  (oo(o)(o))
                                (oo(o(o)))
                                (o(o)(o)(o))
                                (o(o(o)(o)))
                                (o(o(o(o))))
                                (oo((o)(o)))
                                ((o)(o)(o)(o))
                                ((o(o))(o(o)))
                                ((oo)((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))))

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 local disjointness gives A050381.
The non-semi version is A316697.
The same trees counted by number of vertices are A331872.
The Matula-Goebel numbers of these trees are A331873.
Lone-child-avoiding rooted trees counted  by leaves are A000669.
Semi-lone-child-avoiding rooted trees counted by vertices are A331934.
Cf. A000081, A001678, A300660, A316473, A316495, A316696, A331678, A331679, A331680, A331682, A331687, A331871, A331935.

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

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A111299 Numbers whose Matula tree is a binary tree (i.e., root has degree 2 and all nodes except root and leaves have degree 3).

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A331934 Number of semi-lone-child-avoiding rooted trees with n unlabeled vertices.

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A331873 Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted trees.

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A331936 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex (semi-achirality).

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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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A331963 Matula-Goebel numbers of semi-lone-child-avoiding rooted identity trees.

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