A331966 -id:A331966 - 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

A063895 Start with x, xy; then concatenate each word in turn with all preceding words, getting x xy xxy xxxy xyxxy xxxxy xyxxxy xxyxxxy ...; sequence gives number of words of length n. Also binary trees by degree: x (x,y) (x,(x,y)) (x,(x,(x,y))) ((x,y),(x,(x,y)))...

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 11, 22, 43, 88, 179, 372, 774, 1631, 3448, 7347, 15713, 33791, 72923, 158021, 343495, 749102, 1638103, 3591724, 7893802, 17387931, 38379200, 84875596, 188036830, 417284181, 927469845, 2064465341, 4601670625, 10270463565, 22950838755

Offset: 1

Claude Lenormand (claude.lenormand(AT)free.fr), Aug 29 2001

Also binary rooted identity trees (those with no symmetries, cf. A004111).
From Gus Wiseman, May 04 2021: (Start)
Also the number of unlabeled binary rooted semi-identity trees with 2*n - 1 nodes. In a semi-identity tree, only the non-leaf branches directly under any given vertex are required to be distinct. Alternatively, an unlabeled rooted tree is a semi-identity tree iff the non-leaf branches of the root are all distinct and are themselves semi-identity trees. For example, the a(3) = 1 through a(6) = 6 trees are:
  (o(oo))  (o(o(oo)))  ((oo)(o(oo)))  ((oo)(o(o(oo))))  ((o(oo))(o(o(oo))))
                       (o(o(o(oo))))  (o((oo)(o(oo))))  ((oo)((oo)(o(oo))))
                                      (o(o(o(o(oo)))))  ((oo)(o(o(o(oo)))))
                                                        (o((oo)(o(o(oo)))))
                                                        (o(o((oo)(o(oo)))))
                                                        (o(o(o(o(o(oo))))))
The a(8) = 11 trees with 15 nodes:
  ((o(oo))((oo)(o(oo))))
  ((o(oo))(o(o(o(oo)))))
  ((oo)((oo)(o(o(oo)))))
  ((oo)(o((oo)(o(oo)))))
  ((oo)(o(o(o(o(oo))))))
  (o((o(oo))(o(o(oo)))))
  (o((oo)((oo)(o(oo)))))
  (o((oo)(o(o(o(oo))))))
  (o(o((oo)(o(o(oo))))))
  (o(o(o((oo)(o(oo))))))
  (o(o(o(o(o(o(oo)))))))
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for sequences related to rooted trees

Cf. A063894, A036774.
The non-semi-identity version is 2*A001190(n)-1, ranked by A111299.
Semi-binary trees are also counted by A001190, but ranked by A292050.
The not necessarily binary version is A306200, ranked A306202.
The Matula-Goebel numbers of these trees are A339193.
The plane tree version is A343663.
A000081 counts unlabeled rooted trees with n nodes.
A004111 counts identity trees, ranked by A276625.
A306201 counts balanced semi-identity trees, ranked by A306203.
A331966 counts lone-child avoiding semi-identity trees, ranked by A331965.
Cf. A001678, A331934, A331963, A331964.

a:= proc(n) option remember; `if`(n<3, n*(3-n)/2, add(a(i)*a(n-i),
      i=1..(n-1)/2)+`if`(irem(n, 2, 'r')=0, (p->(p-1)*p/2)(a(r)), 0))
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 02 2013

a[n_] := a[n] = If[n<3, n*(3-n)/2, Sum[a[i]*a[n-i], {i, 1, (n-1)/2}]+If[{q, r} = QuotientRemainder[n, 2]; r == 0, (a[q]-1)*a[q]/2, 0]]; Table[a[n], {n, 1, 36}] (* Jean-François Alcover, Feb 25 2014, after Alois P. Heinz *)
ursiq[n_]:=Join@@Table[Select[Union[Sort/@Tuples[ursiq/@ptn]],#=={}||#=={{},{}}||Length[#]==2&&(UnsameQ@@DeleteCases[#,{}])&],{ptn,IntegerPartitions[n-1]}];Table[Length[ursiq[n]],{n,1,15,2}] (* Gus Wiseman, May 04 2021 *)

{a(n)=local(A, m); if(n<1, 0, m=1; A=O(x); while( m<=n, m*=2; A=1-sqrt(1-2*x-2*x^2+subst(A, x, x^2))); polcoeff(A, n))}

a(n) = (sum a(i)*a(j), i+j=n, i2. a(1)=a(2)=1.
G.f. A(x) = 1-sqrt(1-2x-2x^2+A(x^2)) satisfies x+x^2-A(x)+(A(x)^2-A(x^2))/2=0, A(0)=0. - Michael Somos, Sep 06 2003
a(n) ~ c * d^n / n^(3/2), where d = 2.33141659246516873904600076533362924695..., c = 0.2873051160895040470174351963... . - Vaclav Kotesovec, Sep 11 2014

Additional comments and g.f. from Christian G. Bower, Nov 29 2001

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

A331965 Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees.

Original entry on oeis.org

1, 4, 8, 14, 16, 28, 32, 38, 56, 64, 76, 86, 106, 112, 128, 133, 152, 172, 212, 214, 224, 256, 262, 266, 301, 304, 326, 344, 371, 424, 428, 448, 512, 524, 526, 532, 602, 608, 622, 652, 688, 742, 749, 766, 817, 848, 856, 886, 896, 917, 1007, 1024, 1048, 1052

Offset: 1

Gus Wiseman, Feb 04 2020

nonn

First differs from A331683 in having 133, the Matula-Goebel number of the tree ((oo)(ooo)).
Lone-child-avoiding means there are no unary branchings.
In a semi-identity tree, the non-leaf branches of 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, and all composite numbers that are n times a power of two, where n is a squarefree number whose prime indices already belong to the sequence, and a prime index of n is a number m such that prime(m) divides n. [Clarified by Peter Munn and Gus Wiseman, Jun 24 2021]

			The sequence of all lone-child-avoiding rooted semi-identity trees together with their Matula-Goebel numbers begins:
    1: o
    4: (oo)
    8: (ooo)
   14: (o(oo))
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   38: (o(ooo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  133: ((oo)(ooo))
  152: (ooo(ooo))
  172: (oo(o(oo)))
  212: (oo(oooo))
  214: (o(oo(oo)))
The sequence of terms together with their prime indices begins:
    1: {}                 224: {1,1,1,1,1,4}
    4: {1,1}              256: {1,1,1,1,1,1,1,1}
    8: {1,1,1}            262: {1,32}
   14: {1,4}              266: {1,4,8}
   16: {1,1,1,1}          301: {4,14}
   28: {1,1,4}            304: {1,1,1,1,8}
   32: {1,1,1,1,1}        326: {1,38}
   38: {1,8}              344: {1,1,1,14}
   56: {1,1,1,4}          371: {4,16}
   64: {1,1,1,1,1,1}      424: {1,1,1,16}
   76: {1,1,8}            428: {1,1,28}
   86: {1,14}             448: {1,1,1,1,1,1,4}
  106: {1,16}             512: {1,1,1,1,1,1,1,1,1}
  112: {1,1,1,1,4}        524: {1,1,32}
  128: {1,1,1,1,1,1,1}    526: {1,56}
  133: {4,8}              532: {1,1,4,8}
  152: {1,1,1,8}          602: {1,4,14}
  172: {1,1,14}           608: {1,1,1,1,1,8}
  212: {1,1,16}           622: {1,64}
  214: {1,28}             652: {1,1,38}

The non-semi case is {1}.
Not requiring lone-child-avoidance gives A306202.
The locally disjoint version is A331683.
These trees are counted by A331966.
The semi-lone-child-avoiding case is A331994.
Matula-Goebel numbers of rooted identity trees are A276625.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Semi-identity trees are counted by A306200.
Cf. A001678, A004111, A007097, A061775, A122132, A196050, A300660, A316694, A320269, A331681, A331686, A331875, A331936, A331937, A331993.

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

Intersection of A291636 and A306202.

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

A320269 Matula-Goebel numbers of lone-child-avoiding rooted trees in which the non-leaf branches directly under any given node are all equal (semi-achirality).

Original entry on oeis.org

1, 4, 8, 14, 16, 28, 32, 38, 49, 56, 64, 76, 86, 98, 106, 112, 128, 152, 172, 196, 212, 214, 224, 256, 262, 304, 326, 343, 344, 361, 392, 424, 428, 448, 454, 512, 524, 526, 608, 622, 652, 686, 688, 722, 766, 784, 848, 856, 886, 896, 908, 1024, 1042, 1048, 1052

Offset: 1

Gus Wiseman, Oct 08 2018

nonn

First differs from A331871 in lacking 1589.
Lone-child-avoiding means there are no unary branchings.
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.

			The sequence of rooted trees together with their Matula-Goebel numbers begins:
    1: o
    4: (oo)
    8: (ooo)
   14: (o(oo))
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   38: (o(ooo))
   49: ((oo)(oo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
   98: (o(oo)(oo))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  152: (ooo(ooo))
  172: (oo(o(oo)))
  196: (oo(oo)(oo))

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

Cf. A002541, A070776, A167865, A214577, A317710, A320222, A320226.
The same-tree version is A291441.
Not requiring lone-child-avoidance gives A320230.
The enumeration of these trees by vertices is A320268.
The semi-lone-child-avoiding version is A331936.
If the non-leaf branches are all different instead of equal we get A331965.
The fully-achiral case is A331967.
Achiral rooted trees are counted by A003238.
MG-numbers of lone-child-avoiding rooted trees are A291636.
Cf. A061775, A196050, A276625, A280996, A306202, A331912, A331916, A331966.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
hmakQ[n_]:=And[!PrimeQ[n],SameQ@@DeleteCases[primeMS[n],1],And@@hmakQ/@primeMS[n]];Select[Range[1000],hmakQ[#]&]

Updated with corrected terminology by Gus Wiseman, Feb 06 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

A331967 Matula-Goebel numbers of lone-child-avoiding achiral rooted trees.

Original entry on oeis.org

1, 4, 8, 16, 32, 49, 64, 128, 256, 343, 361, 512, 1024, 2048, 2401, 2809, 4096, 6859, 8192, 16384, 16807, 17161, 32768, 51529, 65536, 96721, 117649, 130321, 131072, 148877, 262144, 516961, 524288, 823543, 1048576, 2097152, 2248091, 2476099, 2621161, 4194304

Offset: 1

Gus Wiseman, Feb 06 2020

nonn

Lone-child-avoiding means there are no unary branchings.
In an achiral rooted tree, the branches of any given vertex are all equal.
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 and all numbers of the form prime(j)^k where k > 1 and j is already in the sequence.

			The sequence of all lone-child-avoiding achiral rooted trees together with their Matula-Goebel numbers begins:
      1: o
      4: (oo)
      8: (ooo)
     16: (oooo)
     32: (ooooo)
     49: ((oo)(oo))
     64: (oooooo)
    128: (ooooooo)
    256: (oooooooo)
    343: ((oo)(oo)(oo))
    361: ((ooo)(ooo))
    512: (ooooooooo)
   1024: (oooooooooo)
   2048: (ooooooooooo)
   2401: ((oo)(oo)(oo)(oo))
   2809: ((oooo)(oooo))
   4096: (oooooooooooo)
   6859: ((ooo)(ooo)(ooo))
   8192: (ooooooooooooo)
  16384: (oooooooooooooo)
  16807: ((oo)(oo)(oo)(oo)(oo))
  17161: ((ooooo)(ooooo))
  32768: (ooooooooooooooo)
  51529: (((oo)(oo))((oo)(oo)))
  65536: (oooooooooooooooo)
  96721: ((oooooo)(oooooo))

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
Gus Wiseman, The first 30 lone-child-avoiding achiral rooted trees.

A subset of A025475 (nonprime prime powers).
The enumeration of these trees by vertices is A167865.
Not requiring lone-child-avoidance gives A214577.
The semi-achiral version is A320269.
The semi-lone-child-avoiding version is A331992.
Achiral rooted trees are counted by A003238.
MG-numbers of planted achiral rooted trees are A280996.
MG-numbers of lone-child-avoiding rooted trees are A291636.
Cf. A001678, A007097, A061775, A196050, A276625, A291441, A320230, A320268, A331912, A331936, A331963, A331965, A331966, A331991.

msQ[n_]:=n==1||!PrimeQ[n]&&PrimePowerQ[n]&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[10000],msQ]

Intersection of A214577 (achiral) and A291636 (lone-child-avoiding).

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

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).

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A063895 Start with x, xy; then concatenate each word in turn with all preceding words, getting x xy xxy xxxy xyxxy xxxxy xyxxxy xxyxxxy ...; sequence gives number of words of length n. Also binary trees by degree: x (x,y) (x,(x,y)) (x,(x,(x,y))) ((x,y),(x,(x,y)))...

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

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A331965 Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees.

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

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A320269 Matula-Goebel numbers of lone-child-avoiding rooted trees in which the non-leaf branches directly under any given node are all equal (semi-achirality).

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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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