A306202 -id:A306202 - cp's OEIS frontend

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

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

Offset: 1

Gus Wiseman, Jan 29 2019

nonn

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

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

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

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

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 27 2020

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Also Matula-Goebel numbers of locally disjoint rooted semi-identity trees. Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex. A rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees. The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches (of the root), which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The sequence of all locally disjoint rooted semi-identity trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   6: (o(o))
   7: ((oo))
   8: (ooo)
  10: (o((o)))
  11: ((((o))))
  12: (oo(o))
  13: ((o(o)))
  14: (o(oo))
  15: ((o)((o)))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  20: (oo((o)))
  22: (o(((o))))
  24: (ooo(o))

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

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

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 05 2020

nonn

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

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

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

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

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

Intersection of A306202 and A331935.

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]

A306269 Regular triangle read by rows where T(n,k) is the number of unlabeled balanced rooted semi-identity trees with n >= 1 nodes and depth 0 <= k < n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 2, 2, 1, 1, 1, 0, 1, 3, 3, 2, 1, 1, 1, 0, 1, 3, 4, 3, 2, 1, 1, 1, 0, 1, 5, 6, 5, 3, 2, 1, 1, 1, 0, 1, 5, 9, 7, 5, 3, 2, 1, 1, 1, 0, 1, 7, 12, 12, 8, 5, 3, 2, 1, 1, 1, 0, 1, 8, 17, 17, 13, 8, 5, 3, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Feb 01 2019

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

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  1  1  1
  0  1  2  1  1  1
  0  1  2  2  1  1  1
  0  1  3  3  2  1  1  1
  0  1  3  4  3  2  1  1  1
  0  1  5  6  5  3  2  1  1  1
  0  1  5  9  7  5  3  2  1  1  1
  0  1  7 12 12  8  5  3  2  1  1  1
  0  1  8 17 17 13  8  5  3  2  1  1  1
  0  1 10 25 26 20 14  8  5  3  2  1  1  1
  0  1 12 34 39 31 21 14  8  5  3  2  1  1  1
The postpositive terms of row 9 {3, 4, 3, 2} count the following trees:
  ((ooooooo))   (((oooooo)))    ((((ooooo))))    (((((oooo)))))
  ((o)(ooooo))  (((o)(oooo)))   ((((o)(ooo))))   (((((o)(oo)))))
  ((oo)(oooo))  (((oo)(ooo)))   ((((o))((oo))))
                (((o))((ooo)))

Alois P. Heinz, Rows n = 1..200, flattened
Gus Wiseman, Unlabeled balanced rooted semi-identity trees with 12 nodes, organized by depth.

Row sums give A306201.
T(2n-1,n) gives A306274.
Cf. A000081, A004111, A048816, A079500, A120803, A184155, A276625, A306200, A306202, A306203, A320222, A320270.

ubk[n_,k_]:=Select[Join@@Table[Select[Union[Sort/@Tuples[ubk[#,k-1]&/@ptn]],UnsameQ@@DeleteCases[#,{}]&],{ptn,IntegerPartitions[n-1]}],SameQ[k,##]&@@Length/@Position[#,{}]&];
Table[Length[ubk[n,k]],{n,1,10},{k,0,n-1}]

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

A343937 Number of unlabeled semi-identity plane trees with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 13, 38, 117, 375, 1224, 4095, 13925, 48006, 167259, 588189, 2084948, 7442125, 26725125, 96485782, 350002509, 1275061385, 4662936808, 17111964241, 62996437297, 232589316700, 861028450579, 3195272504259, 11884475937910, 44295733523881, 165420418500155

Offset: 1

Gus Wiseman, May 07 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 iff the non-leaf branches of the root are all distinct and are themselves semi-identity trees.

			The a(1) = 1 through a(5) = 13 trees are the following. The number of nodes is the number of o's plus the number of brackets (...).
  o  (o)  (oo)   (ooo)    (oooo)
          ((o))  ((o)o)   ((o)oo)
                 ((oo))   ((oo)o)
                 (o(o))   ((ooo))
                 (((o)))  (o(o)o)
                          (o(oo))
                          (oo(o))
                          (((o))o)
                          (((o)o))
                          (((oo)))
                          ((o(o)))
                          (o((o)))
                          ((((o))))

Andrew Howroyd, Table of n, a(n) for n = 1..200

The not necessarily semi-identity version is A000108.
The non-plane binary version is A063895, ranked by A339193.
The non-plane version is A306200, ranked by A306202.
The binary case is A343663.
A000081 counts unlabeled rooted trees with n nodes.
A001190*2 - 1 counts binary trees, ranked by A111299.
A001190 counts semi-binary trees, ranked by A292050.
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, A320230, A331934, A331963, A331964.

arsiq[n_]:=Join@@Table[Select[Union[Tuples[arsiq/@ptn]],#=={}||(UnsameQ@@DeleteCases[#,{}])&],{ptn,Join@@Permutations/@IntegerPartitions[n-1]}];
Table[Length[arsiq[n]],{n,10}]

F(p)={my(n=serprec(p,x)-1, q=exp(x*y + O(x*x^n))*prod(k=2, n, (1 + y*x^k + O(x*x^n))^polcoef(p,k,x)) ); sum(k=0, n, k!*polcoef(q,k,y))}
seq(n)={my(p=O(x)); for(n=1, n, p=x*F(p)); Vec(p)} \\ Andrew Howroyd, May 08 2021

G.f.: A(x) satisfies A(x) = x*Sum_{j>=0} j!*[y^j] exp(x*y - Sum_{k>=1} (-y)^k*(A(x^k) - x^k)/k). - Andrew Howroyd, May 08 2021

Terms a(17) and beyond from Andrew Howroyd, May 08 2021

cp's OEIS Frontend

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

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

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

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

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A306269 Regular triangle read by rows where T(n,k) is the number of unlabeled balanced rooted semi-identity trees with n >= 1 nodes and depth 0 <= k < n.

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

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A343937 Number of unlabeled semi-identity plane trees with n nodes.

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