A331871 -id:A331871 - cp's OEIS frontend

A331683 One and all numbers of the form 2^k * prime(j) for k > 0 and j already in the sequence.

1, 4, 8, 14, 16, 28, 32, 38, 56, 64, 76, 86, 106, 112, 128, 152, 172, 212, 214, 224, 256, 262, 304, 326, 344, 424, 428, 448, 512, 524, 526, 608, 622, 652, 688, 766, 848, 856, 886, 896, 1024, 1048, 1052, 1154, 1216, 1226, 1244, 1304, 1376, 1438, 1532, 1696

Offset: 1

Gus Wiseman, Jan 30 2020

nonn

Also Matula-Goebel numbers of lone-child-avoiding rooted trees at with at most one non-leaf branch under any given vertex. A rooted tree is lone-child-avoiding if 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 the root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

Also Matula-Goebel numbers of lone-child-avoiding locally disjoint semi-identity trees. 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 sequence of all 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
    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)
  152: (ooo(ooo))
  172: (oo(o(oo)))
  212: (oo(oooo))
  214: (o(oo(oo)))
  224: (ooooo(oo))

Robert Israel, Table of n, a(n) for n = 1..3140

These trees counted by number of vertices are A212804.
The semi-lone-child-avoiding version is A331681.
The non-semi-identity version is A331871.
Lone-child-avoiding rooted trees are counted by A001678.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Unlabeled semi-identity trees are counted by A306200, with Matula-Goebel numbers A306202.
Locally disjoint rooted trees are counted by A316473.
Matula-Goebel numbers of locally disjoint rooted trees are A316495.
Lone-child-avoiding locally disjoint rooted trees by leaves are A316697.
Cf. A000081, A007097, A061775, A196050, A276625, A316470, A316696, A331678, A331679, A331680, A331682, A331686, A331687.

N:= 10^4: # for terms <= N
S:= {1}:
with(queue):
Q:= new(1):
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,k_}}/;k>1)];
Select[Range[100],uryQ]

Intersection of A291636, A316495, and A306202.

A331935 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 14, 16, 18, 21, 24, 26, 27, 28, 32, 36, 38, 39, 42, 46, 48, 49, 52, 54, 56, 57, 63, 64, 69, 72, 74, 76, 78, 81, 84, 86, 91, 92, 96, 98, 104, 106, 108, 111, 112, 114, 117, 122, 126, 128, 129, 133, 138, 144, 146, 147, 148, 152, 156, 159

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 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 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 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))
  21: ((o)(oo))
  24: (ooo(o))
  26: (o(o(o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  39: ((o)(o(o)))
  42: (o(o)(oo))
The sequence of terms together with their prime indices begins:
    1: {}              46: {1,9}             98: {1,4,4}
    2: {1}             48: {1,1,1,1,2}      104: {1,1,1,6}
    4: {1,1}           49: {4,4}            106: {1,16}
    6: {1,2}           52: {1,1,6}          108: {1,1,2,2,2}
    8: {1,1,1}         54: {1,2,2,2}        111: {2,12}
    9: {2,2}           56: {1,1,1,4}        112: {1,1,1,1,4}
   12: {1,1,2}         57: {2,8}            114: {1,2,8}
   14: {1,4}           63: {2,2,4}          117: {2,2,6}
   16: {1,1,1,1}       64: {1,1,1,1,1,1}    122: {1,18}
   18: {1,2,2}         69: {2,9}            126: {1,2,2,4}
   21: {2,4}           72: {1,1,1,2,2}      128: {1,1,1,1,1,1,1}
   24: {1,1,1,2}       74: {1,12}           129: {2,14}
   26: {1,6}           76: {1,1,8}          133: {4,8}
   27: {2,2,2}         78: {1,2,6}          138: {1,2,9}
   28: {1,1,4}         81: {2,2,2,2}        144: {1,1,1,1,2,2}
   32: {1,1,1,1,1}     84: {1,1,2,4}        146: {1,21}
   36: {1,1,2,2}       86: {1,14}           147: {2,4,4}
   38: {1,8}           91: {4,6}            148: {1,1,12}
   39: {2,6}           92: {1,1,9}          152: {1,1,1,8}
   42: {1,2,4}         96: {1,1,1,1,1,2}    156: {1,1,2,6}

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 enumeration of these trees by leaves is A050381.
The locally disjoint version A331873.
The enumeration of these trees by nodes is A331934.
The case with at most one distinct non-leaf branch of any vertex is A331936.
Lone-child-avoiding rooted trees are counted by A001678.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Cf. A000081, A000669, A007097, A061775, A196050, A320269, A330951, A331871, A331872, A331874, A331937, A331965.

mseQ[n_]:=n==1||n==2||!PrimeQ[n]&&And@@mseQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],mseQ]

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]

A316696 Number of lone-child-avoiding locally disjoint rooted trees whose leaves form an integer partition of n.

Original entry on oeis.org

1, 2, 4, 11, 27, 80, 218, 654, 1923, 5924, 18310, 58176, 186341, 606814, 1993420, 6618160, 22134640

Offset: 1

Gus Wiseman, Jul 10 2018

A rooted tree is lone-child-avoiding if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other (unequal) branch of the same root.

			The a(4) = 11 rooted trees:
  4,
  (13),
  (22),
  (1(12)), (2(11)), (112),
  (1(1(11))), (1(111)), ((11)(11)), (11(11)), (1111).

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

Matula-Goebel numbers of locally disjoint rooted trees are A316495.
The case where all leaves are 1's is A316697.
Lone-child-avoiding locally disjoint rooted trees are A331680.
Cf. A000669, A001678, A141268, A316473, A316652, A331678, A331686, A331687, A331871, A331872, A331874.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],disjointQ],{ptn,Rest[IntegerPartitions[n]]}],{n}];
Table[Length[nms[n]],{n,10}]

a(16)-a(17) from Robert Price, Sep 16 2018

Terminology corrected by Gus Wiseman, Feb 06 2020

A331680 Number of lone-child-avoiding locally disjoint unlabeled rooted trees with n vertices.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 9, 16, 26, 45, 72, 124, 201, 341, 561, 947, 1571, 2651, 4434, 7496, 12631, 21423, 36332, 61910, 105641, 180924, 310548, 534713, 923047

Offset: 1

Gus Wiseman, Jan 25 2020

First differs from A320268 at a(11) = 45, A320268(11) = 44.

A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex. Lone-child-avoiding means there are no unary branchings.

			The a(1) = 1 through a(9) = 16 trees (empty column indicated by dot):
  o  .  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)    (oooooooo)
                     (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))   (o(oooooo))
                              (oo(oo))  (oo(ooo))   (oo(oooo))   (oo(ooooo))
                                        (ooo(oo))   (ooo(ooo))   (ooo(oooo))
                                        ((oo)(oo))  (oooo(oo))   (oooo(ooo))
                                        (o(o(oo)))  (o(o(ooo)))  (ooooo(oo))
                                                    (o(oo)(oo))  ((ooo)(ooo))
                                                    (o(oo(oo)))  (o(o(oooo)))
                                                    (oo(o(oo)))  (o(oo(ooo)))
                                                                 (o(ooo(oo)))
                                                                 (oo(o(ooo)))
                                                                 (oo(oo)(oo))
                                                                 (oo(oo(oo)))
                                                                 (ooo(o(oo)))
                                                                 (o((oo)(oo)))
                                                                 (o(o(o(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.

The enriched version is A316696.
The Matula-Goebel numbers of these trees are A331871.
The non-locally disjoint version is A001678.
These trees counted by number of leaves are A316697.
The semi-lone-child-avoiding version is A331872.
Cf. A000081, A000669, A005804, A060356, A141268, A300660, A316471, A316473, A316694, A316495, A319312, A330465, A331679, A331681, A331683.

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

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

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}]

A331687 Number of locally disjoint enriched p-trees of weight n.

Original entry on oeis.org

1, 2, 4, 12, 29, 93, 249, 803, 2337, 7480, 23130, 77372, 247598, 834507, 2762222

Offset: 1

Gus Wiseman, Jan 31 2020

A locally disjoint enriched p-tree of weight n is either the number n itself or a finite sequence of non-overlapping locally disjoint enriched p-trees whose weights are weakly decreasing and sum to n.

			The a(1) = 1 through a(4) = 12 enriched p-trees:
  1  2     3        4
     (11)  (21)     (22)
           (111)    (31)
           ((11)1)  (211)
                    (1111)
                    ((11)2)
                    ((21)1)
                    (2(11))
                    ((11)11)
                    ((111)1)
                    (((11)1)1)
                    ((11)(11))

The orderless version is A316696.
The identity case is A331684.
P-trees are A196545.
Enriched p-trees are A289501.
Locally disjoint identity trees are A316471.
Enriched identity p-trees are A331875.
Cf. A000669, A141268, A316473, A316495, A316694, A316697, A319312, A331678, A331679, A331680, A331686, A331871, A331874.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
ldep[n_]:=Prepend[Select[Join@@Table[Tuples[ldep/@p],{p,Rest[IntegerPartitions[n]]}],disjointQ[DeleteCases[#,_Integer]]&],n];
Table[Length[ldep[n]],{n,10}]

cp's OEIS Frontend

A331683 One and all numbers of the form 2^k * prime(j) for k > 0 and j already in the sequence.

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A331935 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees.

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

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A316696 Number of lone-child-avoiding locally disjoint rooted trees whose leaves form an integer partition of n.

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A331680 Number of lone-child-avoiding locally disjoint unlabeled rooted 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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A331872 Number of semi-lone-child-avoiding locally disjoint rooted trees with n vertices.

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

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