A111299 -id:A111299 - cp's OEIS frontend

A303026 Matula-Goebel numbers of series-reduced anti-binary (no unary or binary branchings) rooted trees.

1, 8, 16, 32, 64, 76, 128, 152, 212, 256, 304, 424, 512, 524, 608, 722, 848, 1024, 1048, 1216, 1244, 1444, 1532, 1696, 2014, 2048, 2096, 2432, 2488, 2876, 2888, 3064, 3392, 3524, 4028, 4096, 4192, 4864, 4976, 4978, 5204, 5618, 5752, 5776, 6128, 6476, 6784

Offset: 1

Gus Wiseman, Aug 15 2018

nonn

			The sequence of series-reduced anti-binary rooted trees together with their Matula-Goebel numbers begins:
     1: o
     8: (ooo)
    16: (oooo)
    32: (ooooo)
    64: (oooooo)
    76: (oo(ooo))
   128: (ooooooo)
   152: (ooo(ooo))
   212: (oo(oooo))
   256: (oooooooo)
   304: (oooo(ooo))
   424: (ooo(oooo))
   512: (ooooooooo)
   524: (oo(ooooo))
   608: (ooooo(ooo))
   722: (o(ooo)(ooo))
   848: (oooo(oooo))
  1024: (oooooooooo)
  1048: (ooo(ooooo))
  1216: (oooooo(ooo))
  1244: (oo(oooooo))
  1444: (oo(ooo)(ooo))
  1532: (oo(oo(ooo)))
  1696: (ooooo(oooo))
  2014: (o(ooo)(oooo))
  2048: (ooooooooooo)

Cf. A000081, A000598, A001190, A001678, A007097, A061775, A102403, A111299, A292050, A298422, A298424.

Cf. A303022, A303023, A303024, A303025, A303027.

azQ[n_]:=Or[n==1,And[PrimeOmega[n]>2,And@@Cases[FactorInteger[n],{p_,_}:>azQ[PrimePi[p]]]]]
Select[Range[1000],azQ]

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 08 2018

nonn

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

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

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

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

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

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]

A298126 Matula-Goebel numbers of rooted trees in which all outdegrees are even.

Original entry on oeis.org

1, 4, 14, 16, 49, 56, 64, 86, 106, 196, 224, 256, 301, 344, 371, 424, 454, 526, 622, 686, 784, 886, 896, 1024, 1154, 1204, 1376, 1484, 1589, 1696, 1816, 1841, 1849, 2104, 2177, 2279, 2386, 2401, 2488, 2744, 2809, 2846, 3101, 3136, 3238, 3544, 3584, 3986, 4039

Offset: 1

Gus Wiseman, Jan 13 2018

nonn

			Sequence of trees begins:
1   o
4   (oo)
14  (o(oo))
16  (oooo)
49  ((oo)(oo))
56  (ooo(oo))
64  (oooooo)
86  (o(o(oo)))
106 (o(oooo))
196 (oo(oo)(oo))
224 (ooooo(oo))
256 (oooooooo)
301 ((oo)(o(oo)))
344 (ooo(o(oo)))
371 ((oo)(oooo))
424 (ooo(oooo))
454 (o((oo)(oo)))
526 (o(ooo(oo)))
622 (o(oooooo))
686 (o(oo)(oo)(oo))
784 (oooo(oo)(oo))
886 (o(o(o(oo))))
896 (ooooooo(oo))

Index entries for sequences related to Matula-Goebel numbers

Cf. A000081, A001190, A007097, A026424, A028260, A061775, A111299, A214577, A245824, A276625, A277098, A290760, A291441, A291442, A291636, A295461, A297571, A298118, A298120.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
etQ[n_]:=Or[n===1,With[{m=primeMS[n]},EvenQ@Length@m&&And@@etQ/@m]];
Select[Range[10000],etQ]

A298303 Matula-Goebel numbers of rooted trees with thinning limbs.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 39, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 55, 56, 58, 60, 62, 63, 64, 65, 66, 69, 70, 72, 75, 76, 77, 78, 80, 81, 84, 86, 87, 88, 90, 91, 92, 93, 94

Offset: 1

Gus Wiseman, Jan 16 2018

nonn

An unlabeled rooted tree has thinning limbs if its outdegrees are weakly decreasing from root to leaves.

Cf. A000081, A007097, A061775, A111299, A124343, A124346, A214577, A276625, A290760, A291636, A298126, A298120, A298304, A298305.

MGtree[n_]:=If[n===1,{},MGtree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
thinQ[t_]:=And@@Cases[t,b_List:>Length[b]>=Max@@Length/@b,{0,Infinity}];
Select[Range[200],thinQ[MGtree[#]]&]

A298534 Matula-Goebel numbers of rooted trees such that every branch of the root has the same number of leaves.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 27, 29, 30, 31, 32, 33, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 53, 54, 55, 59, 60, 61, 62, 64, 66, 67, 71, 72, 73, 75, 79, 80, 81, 83, 88, 89, 90, 91, 93, 96, 97, 99, 100

Offset: 1

Gus Wiseman, Jan 20 2018

nonn

			Sequence of trees begins:
1  o
2  (o)
3  ((o))
4  (oo)
5  (((o)))
6  (o(o))
7  ((oo))
8  (ooo)
9  ((o)(o))
10 (o((o)))
11 ((((o))))
12 (oo(o))
13 ((o(o)))
15 ((o)((o)))
16 (oooo)
17 (((oo)))
18 (o(o)(o))
19 ((ooo))
20 (oo((o)))
22 (o(((o))))
23 (((o)(o)))
24 (ooo(o))
25 (((o))((o)))
27 ((o)(o)(o))
29 ((o((o))))
30 (o(o)((o)))

Cf. A000081, A007097, A061775, A111299, A214577, A276625, A290760, A290822, A291442, A298533, A298536.

nn=2000;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
leafcount[n_]:=If[n===1,1,With[{m=primeMS[n]},If[Length[m]===1,leafcount[First[m]],Total[leafcount/@m]]]];
Select[Range[nn],SameQ@@leafcount/@primeMS[#]&]

A317719 Numbers that are not powerful tree numbers.

Original entry on oeis.org

6, 10, 12, 13, 14, 15, 18, 20, 21, 22, 24, 26, 28, 29, 30, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82, 84, 85, 86, 87, 88, 89, 90, 91

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

A positive integer n is a powerful tree number iff either n = 1 or n is a prime number whose prime index is a powerful tree number, or n is a powerful number (meaning its prime multiplicities are all greater than 1) whose prime indices are all powerful tree numbers. A prime index of n is a number m such that prime(m) divides n.

			The sequence of numbers that are not powerful tree numbers together with their Matula-Goebel trees begins:
   6: (o(o))
  10: (o((o)))
  12: (oo(o))
  13: ((o(o)))
  14: (o(oo))
  15: ((o)((o)))
  18: (o(o)(o))
  20: (oo((o)))
  21: ((o)(oo))
  22: (o(((o))))
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  29: ((o((o))))
  30: (o(o)((o)))

Complement of A318612.
Cf. A000081, A001694, A061775, A111299, A214577, A276625, A277098, A303431.
Cf. A317102, A317707, A317708, A317709, A317710, A317711, A317712, A317717, A317718.

powgoQ[n_]:=Or[n==1,If[PrimeQ[n],powgoQ[PrimePi[n]],And[Min@@FactorInteger[n][[All,2]]>1,And@@powgoQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[100],!powgoQ[#]&]

A298538 Matula-Goebel numbers of rooted trees such that every branch of the root has the same number of nodes.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 175, 179, 181, 187

Offset: 1

Gus Wiseman, Jan 21 2018

nonn

			Sequence of trees begins:
1  o
2  (o)
3  ((o))
4  (oo)
5  (((o)))
7  ((oo))
8  (ooo)
9  ((o)(o))
11 ((((o))))
13 ((o(o)))
16 (oooo)
17 (((oo)))
19 ((ooo))
23 (((o)(o)))
25 (((o))((o)))
27 ((o)(o)(o))
29 ((o((o))))
31 (((((o)))))

Cf. A000081, A007097, A061775, A111299, A214577, A276625, A290760, A291442, A297571, A298478, A298534, A298536, A298537.

nn=500;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
MGweight[n_]:=If[n===1,1,1+Total[MGweight/@primeMS[n]]];
Select[Range[nn],SameQ@@MGweight/@primeMS[#]&]

A343663 Number of unlabeled binary rooted semi-identity plane trees with 2*n - 1 nodes.

Original entry on oeis.org

1, 1, 2, 4, 12, 34, 108, 344, 1136, 3796, 12920, 44442, 154596, 542336, 1917648, 6825464, 24439008, 87962312, 318087216, 1155090092, 4210494616, 15400782912, 56508464736, 207935588586, 767162495940, 2837260332472, 10516827106016, 39063666532784, 145378611426512

Offset: 1

Gus Wiseman, May 05 2021

nonn

In a semi-identity tree, only the non-leaf branches of any given vertex are required to be 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 a(1) = 1 through a(5) = 12 trees:
  o  (oo)  ((oo)o)  (((oo)o)o)  ((((oo)o)o)o)
           (o(oo))  ((o(oo))o)  (((o(oo))o)o)
                    (o((oo)o))  (((oo)o)(oo))
                    (o(o(oo)))  ((o((oo)o))o)
                                ((o(o(oo)))o)
                                ((o(oo))(oo))
                                ((oo)((oo)o))
                                ((oo)(o(oo)))
                                (o(((oo)o)o))
                                (o((o(oo))o))
                                (o(o((oo)o)))
                                (o(o(o(oo))))

Andrew Howroyd, Table of n, a(n) for n = 1..500
Samuele Giraudo, The combinator M and the Mockingbird lattice, arXiv:2204.03586 [math.CO], 2022.
Samuele Giraudo, Mockingbird lattices, Séminaire Lotharingien de Combinatoire XX, Proceedings of the 34th Conf. on Formal Power, Series and Algebraic Combinatorics (Bangalore, India, 2022).

The not necessarily semi-identity version is A000108.
The non-plane version is A063895, ranked by A339193.
The Matula-Goebel numbers in the non-plane case are A339193.
The not-necessarily binary version is A343937.
A000081 counts unlabeled rooted trees with n nodes.
2*A001190 - 1 counts binary trees, ranked by A111299.
A001190 counts semi-binary trees, ranked by A292050.
A004111 counts identity trees, ranked by A276625.
A306200 counts semi-identity trees, ranked by A306202.
A306201 counts balanced semi-identity trees, ranked by A306203.
A331966 counts lone-child avoiding semi-identity trees, ranked by A331965.
Cf. A001678, A320230, A331934, A331963, A331964.

crsiq[n_]:=Join@@Table[Select[Union[Tuples[crsiq/@ptn]],#=={}||#=={{},{}}||Length[#]==2&&(UnsameQ@@DeleteCases[#,{}])&],{ptn,Join@@Permutations/@IntegerPartitions[n-1]}];
Table[Length[crsiq[n]],{n,1,11,2}]
(* Second program: *)
m = 29; p[_] = 1;
Do[p[x_] = 1 + x + x (p[x]^2 - p[x^2]) + O[x]^m // Normal, {m}];
CoefficientList[p[x], x] (* Jean-François Alcover, May 09 2021, after Andrew Howroyd *)

seq(n)={my(p=O(1)); for(n=1, n, p=1 + x + x*(p^2-subst(p,x,x^2))); Vec(p)} \\ Andrew Howroyd, May 07 2021

G.f.: x*A(x) where A(x) satisfies A(x) = 1 + x + x*(A(x)^2 - A(x^2)). - Andrew Howroyd, May 07 2021

Terms a(13) and beyond from Andrew Howroyd, May 07 2021

A298205 Matula-Goebel numbers of rooted trees in which all outdegrees are either 0, 1, or 3.

Original entry on oeis.org

1, 2, 3, 5, 8, 11, 12, 18, 19, 20, 27, 30, 31, 37, 44, 45, 50, 61, 66, 67, 71, 75, 76, 99, 103, 110, 113, 114, 124, 125, 127, 148, 157, 165, 171, 186, 190, 193, 197, 222, 229, 242, 244, 268, 275, 279, 283, 284, 285, 310, 317, 331, 333, 353, 363, 366, 370, 379

Offset: 1

Gus Wiseman, Jan 14 2018

nonn

			Sequence of rooted trees begins:
1  o
2  (o)
3  ((o))
5  (((o)))
8  (ooo)
11 ((((o))))
12 (oo(o))
18 (o(o)(o))
19 ((ooo))
20 (oo((o)))
27 ((o)(o)(o))
30 (o(o)((o)))
31 (((((o)))))
37 ((oo(o)))
44 (oo(((o))))
45 ((o)(o)((o)))
50 (o((o))((o)))

Cf. A000081, A000598, A014591, A026424, A032305, A061775, A111299, A276625, A292050, A295461, A297571, A298126, A298118, A298120, A298204, A298207.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stQ[n_]:=Or[n===1,With[{m=primeMS[n]},MemberQ[{1,3},Length[m]]&&And@@stQ/@m]];
Select[Range[10000],stQ]

cp's OEIS Frontend

A303026 Matula-Goebel numbers of series-reduced anti-binary (no unary or binary branchings) rooted trees.

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

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

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A298126 Matula-Goebel numbers of rooted trees in which all outdegrees are even.

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A298303 Matula-Goebel numbers of rooted trees with thinning limbs.

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A298534 Matula-Goebel numbers of rooted trees such that every branch of the root has the same number of leaves.

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A317719 Numbers that are not powerful tree numbers.

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A298538 Matula-Goebel numbers of rooted trees such that every branch of the root has the same number of nodes.

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A343663 Number of unlabeled binary rooted semi-identity plane trees with 2*n - 1 nodes.

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