cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 15 results. Next

A324764 Number of anti-transitive rooted identity trees with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 9, 20, 41, 89, 196, 443, 987, 2246, 5114, 11757, 27122, 62898, 146392, 342204, 802429, 1887882
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. It is anti-transitive if the branches of the branches of the root are disjoint from the branches of the root.
Also the number of finitary sets S with n brackets where no element of an element of S is also an element of S. For example, the a(8) = 20 finitary sets are (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}}}

Examples

			The a(1) = 1 through a(7) = 9 anti-transitive 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))))))

Links

Crossrefs

Cf. A000081, A004111, A276625, A279861, A290689, A304360, A306844 (non-identity version), A316500, A317787, A318185.
Cf. A324694, A324751, A324756, A324758, A324765, A324767, A324768, A324770, A324839, A324840, A324844.

Programs

  • Mathematica
    idall[n_]:=If[n==1,{{}},Select[Union[Sort/@Join@@(Tuples[idall/@#]&/@IntegerPartitions[n-1])],UnsameQ@@#&]];
Table[Length[Select[idall[n],Intersection[Union@@#,#]=={}&]],{n,10}]

Extensions

a(21)-a(22) from Jinyuan Wang, Jun 20 2020

A324765 Number of recursively anti-transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 26, 52, 119, 266, 618, 1432, 3402, 8093, 19505, 47228, 115244, 282529, 696388, 1723400
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

An unlabeled rooted tree is recursively anti-transitive if no branch of a branch of a terminal subtree is a branch of the same subtree.

Examples

			The a(1) = 1 through a(6) = 11 recursively anti-transitive rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (((o)))  (((oo)))   (((ooo)))
                          ((o)(o))   ((o)(oo))
                          (o((o)))   (o((oo)))
                          ((((o))))  (oo((o)))
                                     ((((oo))))
                                     (((o)(o)))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

Crossrefs

Cf. A000081, A290689, A301700, A304360, A306844, A317787, A318185.
Cf. A324695, A324751, A324756, A324758, A324764, A324766, A324767, A324768.
Cf. A324838, A324840, A324844.

Programs

  • Mathematica
    nallt[n_]:=Select[Union[Sort/@Join@@(Tuples[nallt/@#]&/@IntegerPartitions[n-1])],Intersection[Union@@#,#]=={}&];
Table[Length[nallt[n]],{n,10}]

A324768 Number of fully anti-transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 27, 60, 152, 376, 968, 2492, 6549, 17259, 46000, 123214, 332304, 900406, 2451999, 6703925
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

An unlabeled rooted tree is fully anti-transitive if no proper terminal subtree of any branch of the root is a branch of the root.

Examples

			The a(1) = 1 through a(6) = 11 rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (((o)))  (((oo)))   (((ooo)))
                          ((o)(o))   ((o)(oo))
                          ((o(o)))   ((o(oo)))
                          ((((o))))  ((oo(o)))
                                     ((((oo))))
                                     (((o)(o)))
                                     (((o(o))))
                                     ((o((o))))
                                     (((((o)))))

Crossrefs

Cf. A000081, A279861, A290689, A304360, A306844, A318185.
Cf. A324695, A324751, A324756, A324758, A324763, A324765, A324769, A324770.
Cf. A324838, A324840, A324844, A324846.

Programs

  • Mathematica
    rtall[n_]:=Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])];
Table[Length[Select[rtall[n],Intersection[Union@@Rest[FixedPointList[Union@@#&,#]],#]=={}&]],{n,10}]

Extensions

a(17)-a(20) from Jinyuan Wang, Jun 20 2020

A324838 Number of unlabeled rooted trees with n nodes where the branches of no branch of the root form a submultiset of the branches of the root.

Original entry on oeis.org

1, 0, 1, 2, 5, 10, 28, 64, 169, 422, 1108, 2872, 7627, 20202, 54216, 145867, 395288
Offset: 1

Views

Author

Gus Wiseman, Mar 18 2019

Keywords

Examples

			The a(1) = 1 through a(6) = 10 rooted trees:
  o  ((o))  ((oo))   ((ooo))    ((oooo))
            (((o)))  (((oo)))   (((ooo)))
                     ((o)(o))   ((o)(oo))
                     ((o(o)))   ((o(oo)))
                     ((((o))))  ((oo(o)))
                                ((((oo))))
                                (((o)(o)))
                                (((o(o))))
                                ((o((o))))
                                (((((o)))))

Crossrefs

Cf. A000081, A290689, A304360, A306844, A317787, A318185.
Cf. A324694, A324696, A324704, A324738, A324744, A324758, A324759, A324765, A324768, A324771, A324839, A324840, A324844, A324846.

Programs

  • Mathematica
    submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
rtall[n_]:=Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])];
Table[Length[Select[rtall[n],And@@Table[!submultQ[b,#],{b,#}]&]],{n,10}]

A358453 Number of transitive ordered rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 37, 83, 190, 444, 1051, 2518, 6090, 14852
Offset: 1

Views

Author

Gus Wiseman, Nov 18 2022

Keywords

Comments

We define an unlabeled ordered rooted tree to be transitive if every branch of a branch of the root already appears farther to the left as a branch of the root. An undirected version is A358454.

Examples

			The a(1) = 1 through a(7) = 17 trees:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)    (oooooo)
                (o(o))  (o(o)o)  (o(o)oo)   (o(o)ooo)
                        (o(oo))  (o(oo)o)   (o(oo)oo)
                        (oo(o))  (o(ooo))   (o(ooo)o)
                                 (oo(o)o)   (o(oooo))
                                 (oo(oo))   (oo(o)oo)
                                 (ooo(o))   (oo(oo)o)
                                 (o(o)(o))  (oo(ooo))
                                            (ooo(o)o)
                                            (ooo(oo))
                                            (oooo(o))
                                            (o(o)(o)o)
                                            (o(o)(oo))
                                            (o(o)o(o))
                                            (o(oo)(o))
                                            (oo(o)(o))
                                            (o(o)((o)))

Crossrefs

The unordered version is A290689, ranked by A290822.
The undirected version is A358454, ranked by A358458.
These trees are ranked by A358457.
A000081 counts rooted trees.
A306844 counts anti-transitive rooted trees.
Cf. A318185, A324695, A324751, A324756, A324758, A324764-A324768, A324838, A324840, A324844, A358456.

Programs

  • Mathematica
    aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],Function[t,And@@Table[Complement[t[[k]],Take[t,k]]=={},{k,Length[t]}]]]],{n,10}]

A143363 Number of ordered trees with n edges and having no protected vertices. A protected vertex in an ordered tree is a vertex at least 2 edges away from its leaf descendants.

Original entry on oeis.org

1, 1, 1, 3, 6, 17, 43, 123, 343, 1004, 2938, 8791, 26456, 80597, 247091, 763507, 2372334, 7413119, 23271657, 73376140, 232238350, 737638868, 2350318688, 7510620143, 24064672921, 77294975952, 248832007318, 802737926643
Offset: 0

Views

Author

Emeric Deutsch, Aug 20 2008

Keywords

Comments

The "no protected vertices" condition can be rephrased as "every non-leaf vertex has at least one leaf child". But a(n) is also the number of ordered trees with n edges in which every non-leaf vertex has at most one leaf child. - David Callan, Aug 22 2014
Also the number of locally non-intersecting ordered rooted trees with n edges, meaning every non-leaf subtree has empty intersection. The unordered version is A007562. - Gus Wiseman, Nov 19 2022
a(n) is the number of parking functions of size n-1 avoiding the patterns 123, 132, and 213 . - Lara Pudwell, Apr 10 2023
For n>0, a(n) is the number of ways to place non-intersecting diagonals in convex n+3-gon so as to create no triangles such that none of the dividing diagonals passes through a chosen vertex. (empirical observation) - Muhammed Sefa Saydam, Feb 14 2025 and Aug 05 2025

Examples

			From _Gus Wiseman_, Nov 19 2022: (Start)
The a(0) = 1 through a(4) = 6 trees with at least one leaf directly under any non-leaf node:
  o  (o)  (oo)  (ooo)   (oooo)
                ((o)o)  ((o)oo)
                (o(o))  ((oo)o)
                        (o(o)o)
                        (o(oo))
                        (oo(o))
The a(0) = 1 through a(4) = 6 trees with at most one leaf directly under any node:
  o  (o)  ((o))  ((o)o)   (((o))o)
                 (o(o))   (((o)o))
                 (((o)))  ((o)(o))
                          ((o(o)))
                          (o((o)))
                          ((((o))))
(End)

Links

Crossrefs

Cf. A143362.
For exactly one leaf directly under any node we have A006013.
The unordered version is A007562, ranked by A316470.
Allowing lone children gives A319378.
A000108 counts ordered rooted trees, unordered A000081.
A358453 counts transitive ordered trees, unordered A290689.
A358460 counts locally disjoint ordered trees, unordered A316473.
Cf. A318185, A324756, A324767, A324840, A324844.

Programs

  • Maple
    p:=z^2*G^3-2*z*G^2-2*z^2*G^2+3*z*G+G+z^2*G-1-2*z=0: G:=RootOf(p,G): Gser:= series(G,z=0,33): seq(coeff(Gser,z,n),n=0..28);
  • Mathematica
    a[n_Integer] := a[n] = Round[SeriesCoefficient[2 (x + 1 - Sqrt[x^2 - x + 1] Cos[ArcTan[(3 x Sqrt[12 x^3 - 96 x^2 - 24 x + 15])/(2 x^3 - 30 x^2 - 3 x + 2)]/3])/(3 x), {x, 0, n}]]; Table[a[n], {n, 0, 20}] (* Vladimir Reshetnikov, Apr 10 2022 *)
RecurrenceTable[{25 (n + 5) (n + 6) a[n + 5] - 10 (n + 5) (5 n + 21) a[n + 4] - 2 (77 n^2 + 613 n + 1185) a[n + 3] + 2 (50 n^2 + 253 n + 312) a[n + 2] + 4 (2 n + 1) (7 n + 9) a[n + 1] - 4 n (2 n + 1) a[n] == 0, a[0] == 1, a[1] == 1, a[2] == 1, a[3] == 3, a[4] == 6}, a[n], {n, 0, 27}] (* Vladimir Reshetnikov, Apr 11 2022 *)
ait[n_]:=ait[n]=If[n==1,{{}},Join@@Table[Select[Tuples[ait/@c],MemberQ[#,{}]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[ait[n]],{n,15}] (* Gus Wiseman, Nov 19 2022 *)

Formula

a(n) = A143362(n,0) for n>=1.
G.f.: G=G(z) satisfies z^2*G^3-2z(1+z)G^2+(1+3z+z^2)G-(1+2z)=0.
G.f.: (x+1-sqrt(x^2-x+1)*cos(arctan((3*x*sqrt(12*x^3-96*x^2-24*x+15))/(2*x^3-30*x^2-3*x+2))/3))*2/(3*x). - Vladimir Reshetnikov, Apr 10 2022
Recurrence: 25*(n+5)*(n+6)*a(n+5) - 10*(n+5)*(5*n+21)*a(n+4) - 2*(77*n^2+613*n+1185)*a(n+3) + 2*(50*n^2+253*n+312)*a(n+2) + 4*(2*n+1)*(7*n+9)*a(n+1) - 4*n*(2*n+1)*a(n) = 0. - Vladimir Reshetnikov, Apr 11 2022
From Muhammed Sefa Saydam, Jul 12 2025: (Start)
a(n) = Sum_{k=2..n+2} A046736(k) * A046736(n-k+3) , for n >= 0 and A046736(1) = 1.
a(n) = A049125(n) + Sum_{k=1..n-2} A049125(k) * A046736(n-k+2), for n >= 3.
a(n) = A049125(n) + Sum_{k=1..n-2} a(k) * a(n-k-1), for n >= 3. (End)

A324767 Number of recursively anti-transitive rooted identity trees with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 9, 17, 33, 63, 126, 254, 511, 1039, 2124, 4371, 9059, 18839, 39339, 82385, 173111, 364829, 771010, 1633313
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

An unlabeled rooted tree is recursively anti-transitive if no branch of a branch of any terminal subtree is a branch of the same subtree. It is an identity tree if there are no repeated branches directly under a common root.
Also the number of finitary sets with n brackets where, at any level, no element of an element of a set is an element of the same set. For example, the a(8) = 9 finitary sets are (o = {}):
{{{{{{{o}}}}}}}
{{{{o,{{o}}}}}}
{{{o,{{{o}}}}}}
{{o,{{{{o}}}}}}
{{{o},{{{o}}}}}
{o,{{{{{o}}}}}}
{o,{{o,{{o}}}}}
{{o},{{{{o}}}}}
{{o},{o,{{o}}}}
The Matula-Goebel numbers of these trees are given by A324766.

Examples

			The a(4) = 1 through a(8) = 9 recursively anti-transitive 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)))))))

Crossrefs

Cf. A000081, A004111, A276625, A279861, A290689, A290760, A304360, A306844, A316500.
Cf. A324695, A324751, A324758, A324764 (non-recursive version), A324765 (non-identity version), A324766, A324770, A324839, A324840, A324844.

Programs

  • Mathematica
    iallt[n_]:=Select[Union[Sort/@Join@@(Tuples[iallt/@#]&/@IntegerPartitions[n-1])],UnsameQ@@#&&Intersection[Union@@#,#]=={}&];
Table[Length[iallt[n]],{n,10}]

A324770 Number of fully anti-transitive rooted identity trees with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 13, 27, 58, 128, 286, 640, 1452, 3308, 7594, 17512, 40591, 94449, 220672
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

An unlabeled rooted tree is fully anti-transitive if no proper terminal subtree of any branch of the root is a branch of the root. It is an identity tree if there are no repeated branches directly under the same root.

Examples

			The a(1) = 1 through a(7) = 6 fully anti-transitive 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))))))

Links

Crossrefs

Cf. A000081, A004111, A276625, A279861, A290760, A304360, A306844.
Cf. A324695, A324751, A324763, A324764, A324765, A324767, A324769.
Cf. A324839, A324840, A324843, A324844, A324846.

Programs

  • Mathematica
    idall[n_]:=If[n==1,{{}},Select[Union[Sort/@Join@@(Tuples[idall/@#]&/@IntegerPartitions[n-1])],UnsameQ@@#&]];
Table[Length[Select[idall[n],Intersection[Union@@Rest[FixedPointList[Union@@#&,#]],#]=={}&]],{n,10}]

A358456 Number of recursively bi-anti-transitive ordered rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 7, 17, 47, 117, 321, 895, 2556, 7331, 21435, 63116, 187530
Offset: 1

Views

Author

Gus Wiseman, Nov 18 2022

Keywords

Comments

We define an unlabeled ordered rooted tree to be recursively bi-anti-transitive if there are no two branches of the same node such that one is a branch of the other.

Examples

			The a(1) = 1 through a(6) = 17 trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (((o)))  (((o))o)   (((o))oo)
                          (((oo)))   (((oo))o)
                          ((o)(o))   (((ooo)))
                          (o((o)))   ((o)(oo))
                          ((((o))))  ((oo)(o))
                                     (o((o))o)
                                     (o((oo)))
                                     (oo((o)))
                                     ((((o)))o)
                                     ((((o))o))
                                     ((((oo))))
                                     (((o)(o)))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

Crossrefs

The unordered version is A324765, ranked by A324766.
The directed version is A358455.
A000108 counts ordered rooted trees, unordered A000081.
A306844 counts anti-transitive rooted trees.
A358453 counts transitive ordered trees, unordered A290689.
Cf. A318185, A324695, A324751, A324756, A324758, A324764, A324767, A324768, A324838, A324840, A324844.

Programs

  • Mathematica
    aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],FreeQ[#,{_,x_,_,{_,x_,_},_}|{_,{_,x_,_},_,x_,_}]&]],{n,10}]

A324839 Number of unlabeled rooted identity trees with n nodes where the branches of no branch of the root form a subset of the branches of the root.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 8, 16, 35, 74, 166, 367, 831, 1878, 4299, 9857, 22775, 52777, 122957, 287337
Offset: 1

Views

Author

Gus Wiseman, Mar 18 2019

Keywords

Comments

An unlabeled rooted tree is an identity tree if there are no repeated branches directly under the same root.
Also the number of finitary sets with n brackets where no element is also a subset. For example, the a(7) = 8 sets are (o = {}):
{{{{{{o}}}}}}
{{{{o,{o}}}}}
{{{o,{{o}}}}}
{{o,{{{o}}}}}
{{o,{o,{o}}}}
{{{o},{{o}}}}
{{o},{{{o}}}}
{{o},{o,{o}}}

Examples

			The a(1) = 1 through a(8) = 16 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)))))))

Crossrefs

Cf. A000081, A290760, A304360, A306844, A317787.
Cf. A324694, A324696, A324704, A324738, A324744, A324758, A324759, A324767, A324770, A324771, A324838, A324840, A324844, A324846.

Programs

  • Mathematica
    idall[n_]:=If[n==1,{{}},Select[Union[Sort/@Join@@(Tuples[idall/@#]&/@IntegerPartitions[n-1])],UnsameQ@@#&]];
Table[Length[Select[idall[n],And@@Table[!SubsetQ[#,b],{b,#}]&]],{n,10}]
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