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.

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A358582 Number of rooted trees with n nodes, most of which are not leaves.

Original entry on oeis.org

0, 0, 1, 1, 5, 7, 28, 48, 176, 336, 1179, 2420, 8269, 17855, 59832, 134289, 443407, 1025685, 3346702, 7933161, 25632265, 62000170, 198670299, 488801159, 1555187172, 3882403641, 12276230777, 31034921462, 97601239282, 249471619165, 780790439063, 2015194486878
Offset: 1

Views

Author

Gus Wiseman, Nov 23 2022

Keywords

Examples

			The a(3) = 1 through a(6) = 7 trees:
  ((o))  (((o)))  (((oo)))   ((((oo))))
                  ((o)(o))   (((o)(o)))
                  ((o(o)))   (((o(o))))
                  (o((o)))   ((o)((o)))
                  ((((o))))  ((o((o))))
                             (o(((o))))
                             (((((o)))))

Links

Crossrefs

For equality we have A185650 aerated, ranked by A358578.
The opposite version is A358581, non-strict A358583.
The non-strict version is A358584.
The ordered version is A358585, odd-indexed terms A065097.
A000081 counts rooted trees, ordered A000108.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
A358575 counts rooted trees by nodes and internal nodes, ordered A090181.
A358589 counts square trees, ranked by A358577, ordered A358590.
Cf. A000891, A034781, A109129, A342507, A358579, A358580, A358586, A358591.

Programs

  • Mathematica
    art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,{},{0,Infinity}][_],{0,Infinity}]&]],{n,0,10}]
  • PARI
    \\ See A358584 for R(n).
seq(n) = {my(A=R(n)); vector(n, n, vecsum(Vecrev(A[n]/y)[1..(n-1)\2]))} \\ Andrew Howroyd, Dec 30 2022

Formula

A358581(n) + A358584(n) = A000081(n).
A358582(n) + A358583(n) = A000081(n).
a(n) = Sum_{k=1..floor((n-1)/2)} A055277(n, k). - Andrew Howroyd, Dec 30 2022

Extensions

Terms a(19) and beyond from Andrew Howroyd, Dec 30 2022

A358585 Number of ordered rooted trees with n nodes, most of which are leaves.

Original entry on oeis.org

1, 0, 1, 1, 7, 11, 66, 127, 715, 1549, 8398, 19691, 104006, 258194, 1337220, 3467115, 17678835, 47440745, 238819350, 659060677, 3282060210, 9271024542, 45741281820, 131788178171, 644952073662, 1890110798926, 9183676536076, 27316119923002, 131873975875180, 397407983278484
Offset: 1

Views

Author

Gus Wiseman, Nov 24 2022

Keywords

Examples

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

Links

Crossrefs

For equality we have A000891, unordered A185650.
Odd-indexed terms are A065097.
The unordered version is A358581.
The opposite is the same, unordered A358582.
The non-strict version is A358586, unordered A358583.
A000108 counts ordered rooted trees, unordered A000081.
A001263 counts ordered rooted trees by nodes and leaves, unordered A055277.
A080936 counts ordered rooted trees by nodes and height, unordered A034781.
A090181 counts ordered rooted trees by nodes and internals, unord. A358575.
A358590 counts square ordered trees, unordered A358589 (ranked by A358577).
Cf. A109129, A342507, A358371, A358579, A358580, A358584, A358588, A358590.

Programs

  • Mathematica
    aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],Count[#,{},{0,Infinity}]>Count[#,[_],{0,Infinity}]&]],{n,10}]
  • PARI
    a(n) = if(n==1, 1, n--; (binomial(2*n,n)/(n+1) - if(n%2, binomial(n, (n-1)/2)^2 / n))/2) \\ Andrew Howroyd, Jan 13 2024

Formula

From Andrew Howroyd, Jan 13 2024: (Start)
a(n) = Sum_{k=1..floor((n-1)/2)} A001263(n-1, k) for n >= 2.
a(2*n) = (A000108(2*n-1) - A000891(n-1))/2 for n >= 1;
a(2*n+1) = A000108(2*n)/2 for n >= 1. (End)

Extensions

a(16) onwards from Andrew Howroyd, Jan 13 2024

A358583 Number of rooted trees with n nodes, at least half of which are leaves.

Original entry on oeis.org

1, 1, 1, 3, 4, 13, 20, 67, 110, 383, 663, 2346, 4217, 15118, 27979, 101092, 191440, 695474, 1341974, 4893067, 9589567, 35055011, 69612556, 254923825, 511987473, 1877232869, 3807503552, 13972144807, 28585315026, 104955228432, 216381073935, 794739865822
Offset: 1

Views

Author

Gus Wiseman, Nov 23 2022

Keywords

Examples

			The a(1) = 1 through a(6) = 13 trees:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)
                ((oo))  ((ooo))  ((oooo))
                (o(o))  (o(oo))  (o(ooo))
                        (oo(o))  (oo(oo))
                                 (ooo(o))
                                 (((ooo)))
                                 ((o)(oo))
                                 ((o(oo)))
                                 ((oo(o)))
                                 (o((oo)))
                                 (o(o)(o))
                                 (o(o(o)))
                                 (oo((o)))

Links

Crossrefs

For equality we have A185650 aerated, ranked by A358578.
The strict case is A358581.
The opposite version is A358584, strict A358582.
The ordered version is A358586, strict A358585.
A000081 counts rooted trees, ordered A000108.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
A358575 counts rooted trees by nodes and internal nodes, ordered A090181.
A358589 counts square rooted trees, ranked by A358577, ordered A358590.
Cf. A000891, A034781, A065097, A109129, A342507, A358579, A358580, A358586.

Programs

  • Mathematica
    art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,{},{0,Infinity}]>=Count[#,[_],{0,Infinity}]&]],{n,1,10}]
  • PARI
    \\ See A358584 for R(n).
seq(n) = {my(A=R(n)); vector(n, n, my(u=Vecrev(A[n]/y)); vecsum(u[(n-1)\2+1..#u]))} \\ Andrew Howroyd, Dec 31 2022

Formula

A358581(n) + A358584(n) = A000081(n).
A358582(n) + A358583(n) = A000081(n).
a(n) = Sum_{k=floor((n-1)/2)+1..n} A055277(n, k). - Andrew Howroyd, Dec 31 2022

Extensions

Terms a(19) and beyond from Andrew Howroyd, Dec 31 2022

A358553 Number of internal (non-leaf) nodes in the n-th standard ordered rooted tree.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 2, 1, 2, 3, 3, 2, 3, 2, 2, 1, 4, 2, 4, 3, 4, 3, 3, 2, 2, 3, 3, 2, 3, 2, 2, 1, 3, 4, 3, 2, 5, 4, 4, 3, 3, 4, 4, 3, 4, 3, 3, 2, 4, 2, 4, 3, 4, 3, 3, 2, 2, 3, 3, 2, 3, 2, 2, 1, 3, 3, 5, 4, 4, 3, 3, 2, 4, 5, 5, 4, 5, 4, 4, 3, 5, 3, 5, 4, 5, 4, 4
Offset: 1

Views

Author

Gus Wiseman, Nov 26 2022

Keywords

Comments

We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

Examples

			The 89-th standard rooted tree is ((o)o(oo)), and it has 3 internal nodes, so a(89) = 3.

Crossrefs

This statistic is counted by A001263, unordered A358575 (reverse A055277).
The unordered version is A342507, firsts A358554.
Other statistics: A358371 (leaves), A358372 (nodes), A358379 (edge-height).
A000081 counts rooted trees, ordered A000108.
Cf. A000891, A014221, A358373, A358585, A358586, A358588.

Programs

  • Mathematica
    stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Table[Count[srt[n],[_],{0,Infinity}],{n,100}]

A358723 Number of n-node rooted trees of edge-height equal to their number of leaves.

Original entry on oeis.org

0, 1, 0, 2, 1, 6, 7, 26, 43, 135, 276, 755, 1769, 4648, 11406, 29762, 75284, 195566, 503165, 1310705, 3402317, 8892807, 23231037, 60906456, 159786040, 420144405, 1105673058, 2914252306, 7688019511, 20304253421, 53667498236, 141976081288, 375858854594, 995728192169
Offset: 1

Views

Author

Gus Wiseman, Nov 29 2022

Keywords

Comments

Edge-height (A109082) is the number of edges in the longest path from root to leaf.

Examples

			The a(1) = 0 through a(7) = 7 trees:
  .  (o)  .  ((oo))  ((o)(o))  (((ooo)))  (((o))(oo))
             (o(o))            ((o(oo)))  (((o)(oo)))
                               ((oo(o)))  ((o)((oo)))
                               (o((oo)))  ((o)(o(o)))
                               (o(o(o)))  ((o(o)(o)))
                               (oo((o)))  (o((o)(o)))
                                          (o(o)((o)))

Links

Crossrefs

For internals instead of leaves: A011782, ranked by A209638.
For internals instead of edge-height: A185650 aerated, ranked by A358578.
For node-height: A358589 (square trees), ranked by A358577, ordered A358590.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height, ordered A080936.
A055277 counts rooted trees by nodes and leaves, ordered A001263.
A358575 counts rooted trees by nodes and internals, ordered A090181.
Cf. A065097, A109082, A109129, A342507, A358552, A358587, A358591, A358728.

Programs

  • Mathematica
    art[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[art/@c],OrderedQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[art[n],Count[#,{},{-2}]==Depth[#]-2&]],{n,1,10}]
  • PARI
    \\ Needs R(n,f) defined in A358589.
seq(n) = {Vec(R(n, (h,p)->polcoef(p,h-1,y)), -n)} \\ Andrew Howroyd, Jan 01 2023

Extensions

Terms a(19) and beyond from Andrew Howroyd, Jan 01 2023
Previous Showing 11-15 of 15 results.

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