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 13 results. Next

A124343 Number of rooted trees on n nodes with thinning limbs.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 21, 38, 78, 153, 314, 632, 1313, 2700, 5646, 11786, 24831, 52348, 111027, 235834, 502986, 1074739, 2303146, 4944507, 10639201, 22930493, 49511948, 107065966, 231874164, 502834328, 1091842824, 2373565195, 5165713137, 11254029616, 24542260010
Offset: 1

Views

Author

Christian G. Bower, Oct 30 2006, suggested by Franklin T. Adams-Watters

Keywords

Comments

A rooted tree with thinning limbs is such that if a node has k children, all its children have at most k children.

Examples

			The a(5) = 6 trees are ((((o)))), (o((o))), (o(oo)), ((o)(o)), (oo(o)), (oooo). - _Gus Wiseman_, Jan 25 2018

Links

Crossrefs

Cf. A000081, A032305, A124344-A124348, A290689, A298303, A298304, A298305, A298422.
Row sums of A244657.

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n A(n$2):
seq(a(n), n=1..35);  # Alois P. Heinz, Jul 08 2014
  • Mathematica
    b[n_, i_, h_, v_] := b[n, i, h, v] = If[n==0, If[v==0, 1, 0], If[i<1 || v<1 || nJean-François Alcover, Mar 01 2016, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Jul 04 2014

A245120 Number T(n,k) of n-node rooted identity trees with thinning limbs and root outdegree (branching factor) k; triangle T(n,k), n>=1, 0<=k<=max-index-of-row(n), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 3, 0, 1, 4, 1, 0, 1, 8, 2, 0, 1, 12, 4, 0, 1, 22, 9, 0, 1, 36, 17, 2, 0, 1, 63, 35, 3, 0, 1, 107, 67, 9, 0, 1, 188, 131, 20, 0, 1, 327, 249, 46, 1, 0, 1, 578, 484, 94, 4, 0, 1, 1020, 922, 202, 11, 0, 1, 1820, 1775, 412, 28
Offset: 1

Views

Author

Alois P. Heinz, Jul 12 2014

Keywords

Comments

In a rooted tree with thinning limbs the outdegree of a parent node is larger than or equal to the outdegree of any of its child nodes.

Examples

			The A124346(7) = 6 7-node rooted identity trees with thinning limbs sorted by root outdegree 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  :                            :       :
:     :                            :       :
: -1- : -------------2------------ : --3-- :
Thus row 7 = [0, 1, 4, 1].
Triangle T(n,k) begins:
1;
0, 1;
0, 1;
0, 1,  1;
0, 1,  1;
0, 1,  3;
0, 1,  4,  1;
0, 1,  8,  2;
0, 1, 12,  4;
0, 1, 22,  9;
0, 1, 36, 17, 2;
0, 1, 63, 35, 3;

Links

Crossrefs

Column k=0-10 give: A000007(n-1), A000012 (for n>1), A245121, A245122, A245123, A245124, A245125, A245126, A245127, A245128, A245129.
Row sums give A124346.
Cf. A244657.

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0, `if`(v=0, 1, 0),
      `if`(i<1 or v<1 or n0 do od; k-1 fi
    end:
T:= (n, k)-> b(n-1$2, k$2):
seq(seq(T(n, k), k=0..g(n)), n=1..25);
  • Mathematica
    b[n_, i_, h_, v_] := b[n, i, h, v] = If[n==0, If[v==0, 1, 0], If[i<1 || v<1 || n0, k++]; k-1]; T[n_, k_] := b[n-1, n-1, k, k]; Table[T[n, k], {n, 1, 25}, {k, 0, g[n]}] // Flatten (* Jean-François Alcover, Jan 18 2017, translated from Maple *)

A245151 Number T(n,k) of n-node unlabeled rooted trees with thickening limbs and root outdegree (branching factor) k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 1, 0, 5, 1, 0, 0, 1, 0, 7, 3, 1, 0, 0, 1, 0, 12, 3, 1, 0, 0, 0, 1, 0, 17, 8, 1, 1, 0, 0, 0, 1, 0, 28, 9, 3, 1, 0, 0, 0, 0, 1, 0, 42, 21, 3, 1, 1, 0, 0, 0, 0, 1, 0, 69, 28, 5, 1, 1, 0, 0, 0, 0, 0, 1, 0, 105, 56, 9, 3, 1, 1, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Alois P. Heinz, Jul 12 2014

Keywords

Comments

In a rooted tree with thickening limbs the outdegree of a parent node is smaller than or equal to the outdegree of any of its non-leaf child nodes.
T(n+1,1) = Sum_{k=0..n-1} T(n,k) for n>=1.
T(n+1,n) = T(2n+1,n) = 1 for n>=0.
T(n,1+floor((n-1)/2)) = 0 for n>3.

Examples

			The A245152(5) = 5 5-node rooted trees with thickening limbs sorted by root outdegree 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             :         :         :
:               :         :         :
: ------1------ : ---2--- : ---4--- :
Thus row 5 = [0, 3, 1, 0, 1].
Triangle T(n,k) begins:
1;
0,   1;
0,   1,  1;
0,   2,  0,  1;
0,   3,  1,  0, 1;
0,   5,  1,  0, 0, 1;
0,   7,  3,  1, 0, 0, 1;
0,  12,  3,  1, 0, 0, 0, 1;
0,  17,  8,  1, 1, 0, 0, 0, 1;
0,  28,  9,  3, 1, 0, 0, 0, 0, 1;
0,  42, 21,  3, 1, 1, 0, 0, 0, 0, 1;
0,  69, 28,  5, 1, 1, 0, 0, 0, 0, 0, 1;
0, 105, 56,  9, 3, 1, 1, 0, 0, 0, 0, 0, 1;
0, 176, 81, 12, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1;

Links

Crossrefs

Columns k=0-10 give: A000007(n-1), A245152(n-1), A245142, A245143, A245144, A245145, A245146, A245147, A245148, A245149, A245150.
Row sums give A245152.
Cf. A244657 (thinning limbs).

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n b(n-1$2, k$2):
seq(seq(T(n, k), k=0..n-1), n=1..20);
  • Mathematica
    b[n_, i_, h_, v_] := b[n, i, h, v] = If[n == 0, If[v == 0, 1, 0], If[i<1 || v<1 || nJean-François Alcover, Jan 27 2015, after Alois P. Heinz *)

A244703 Number of n-node unlabeled rooted trees with thinning limbs and root outdegree (branching factor) 2.

Original entry on oeis.org

1, 1, 3, 4, 9, 13, 26, 42, 81, 138, 262, 467, 885, 1620, 3076, 5743, 10953, 20721, 39714, 75873, 146139, 281259, 544230, 1053552, 2047147, 3981790, 7766018, 15165195, 29676887, 58148087, 114129308, 224278526, 441368913, 869583189, 1715365690, 3387344619
Offset: 3

Views

Author

Alois P. Heinz, Jul 04 2014

Keywords

Comments

In a rooted tree with thinning limbs the outdegree of a parent node is larger than or equal to the outdegree of any of its child nodes.

Examples

			a(6) = 4:
    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

Column k=2 of A244657.

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n b(n-1$2, 2$2):
seq(a(n), n=3..50);
  • Mathematica
    b[n_, i_, h_, v_] := b[n, i, h, v] = If[n == 0, If[v == 0, 1, 0], If[i < 1 || v < 1 || n < v, 0, If[n == v, 1, Sum[Binomial[A[i, Min[i - 1, h]] + j - 1, j]*b[n - i*j, i - 1, h, v - j], {j, 0, Min[n/i, v]}]]]];
A[n_, k_] := A[n, k] = If[n<2, n, Sum[b[n-1, n-1, j, j], {j, 1, Min[k, n-1] }]];
a[n_] := b[n-1, n-1, 2, 2];
a /@ Range[3, 50] (* Jean-François Alcover, Dec 27 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 2.0554620926822709065075792..., c = 1.0209036918758320315742... . - Vaclav Kotesovec, Aug 27 2014

A244704 Number of n-node unlabeled rooted trees with thinning limbs and root outdegree (branching factor) 3.

Original entry on oeis.org

1, 1, 3, 6, 13, 25, 55, 107, 224, 454, 938, 1916, 3969, 8163, 16918, 35010, 72724, 151093, 314749, 656115, 1370348, 2864948, 5998547, 12572884, 26385837, 55431031, 116577538, 245415158, 517152607, 1090771973, 2302729115, 4865449045, 10288826434, 21774842539
Offset: 4

Views

Author

Alois P. Heinz, Jul 04 2014

Keywords

Comments

In a rooted tree with thinning limbs the outdegree of a parent node is larger than or equal to the outdegree of any of its child nodes.

Examples

			a(7) = 6:
    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

Links

Crossrefs

Column k=3 of A244657.

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n b(n-1$2, 3$2):
seq(a(n), n=4..50);
  • Mathematica
    b[n_, i_, h_, v_] := b[n, i, h, v] = If[n == 0, If[v == 0, 1, 0], If[i < 1 || v < 1 || n < v, 0, If[n == v, 1, Sum[Binomial[A[i, Min[i - 1, h]] + j - 1, j]*b[n - i*j, i - 1, h, v - j], {j, 0, Min[n/i, v]}]]]];
A[n_, k_] := A[n, k] = If[n < 2, n, Sum[b[n - 1, n - 1, j, j], {j, 1, Min[k, n - 1]}]];
a[n_] := b[n-1, n-1, 3, 3];
a /@ Range[4, 50] (* Jean-François Alcover, Dec 27 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 2.1991393868..., c = 1.0259536... . - Vaclav Kotesovec, Aug 27 2014

A244705 Number of n-node unlabeled rooted trees with thinning limbs and root outdegree (branching factor) 4.

Original entry on oeis.org

1, 1, 3, 6, 15, 29, 68, 140, 312, 660, 1443, 3084, 6710, 14425, 31278, 67508, 146300, 316424, 685955, 1486008, 3223480, 6992012, 15179437, 32960891, 71617874, 155661971, 338508703, 736401503, 1602712182, 3489454243, 7600403101, 16560519877, 36097320801
Offset: 5

Views

Author

Alois P. Heinz, Jul 04 2014

Keywords

Comments

In a rooted tree with thinning limbs the outdegree of a parent node is larger than or equal to the outdegree of any of its child nodes.

Links

Crossrefs

Column k=4 of A244657.

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n b(n-1$2, 4$2):
seq(a(n), n=5..50);

A244706 Number of n-node unlabeled rooted trees with thinning limbs and root outdegree (branching factor) 5.

Original entry on oeis.org

1, 1, 3, 6, 15, 31, 72, 153, 346, 752, 1673, 3661, 8108, 17814, 39349, 86646, 191251, 421596, 930519, 2052789, 4531648, 10002857, 22088709, 48780279, 107757048, 238069894, 526096509, 1162775782, 2570487392, 5683401236, 12568472173, 27799055016, 61496981626
Offset: 6

Views

Author

Alois P. Heinz, Jul 04 2014

Keywords

Comments

In a rooted tree with thinning limbs the outdegree of a parent node is larger than or equal to the outdegree of any of its child nodes.

Links

Crossrefs

Column k=5 of A244657.

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n b(n-1$2, 5$2):
seq(a(n), n=6..50);

A244707 Number of n-node unlabeled rooted trees with thinning limbs and root outdegree (branching factor) 6.

Original entry on oeis.org

1, 1, 3, 6, 15, 31, 74, 157, 359, 786, 1766, 3895, 8710, 19287, 42987, 95437, 212468, 472204, 1050940, 2337221, 5201558, 11573156, 25759514, 57332239, 127633669, 284148877, 632704464, 1408925270, 3137861761, 6989057709, 15568767849, 34684141315, 77277619879
Offset: 7

Views

Author

Alois P. Heinz, Jul 04 2014

Keywords

Comments

In a rooted tree with thinning limbs the outdegree of a parent node is larger than or equal to the outdegree of any of its child nodes.

Links

Crossrefs

Column k=6 of A244657.

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n b(n-1$2, 6$2):
seq(a(n), n=7..50);

A244708 Number of n-node unlabeled rooted trees with thinning limbs and root outdegree (branching factor) 7.

Original entry on oeis.org

1, 1, 3, 6, 15, 31, 74, 159, 363, 799, 1800, 3988, 8945, 19893, 44486, 99153, 221520, 494187, 1103789, 2463834, 5502927, 12288076, 27448039, 61308387, 136966368, 305999360, 683733350, 1527844853, 3414432569, 7631131801, 17056871547, 38127833992, 85235556468
Offset: 8

Views

Author

Alois P. Heinz, Jul 04 2014

Keywords

Comments

In a rooted tree with thinning limbs the outdegree of a parent node is larger than or equal to the outdegree of any of its child nodes.

Links

Crossrefs

Column k=7 of A244657.

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n b(n-1$2, 7$2):
seq(a(n), n=8..50);

A244709 Number of n-node unlabeled rooted trees with thinning limbs and root outdegree (branching factor) 8.

Original entry on oeis.org

1, 1, 3, 6, 15, 31, 74, 159, 365, 803, 1813, 4022, 9038, 20128, 45093, 100656, 225263, 503320, 1126045, 2517487, 5631913, 12596046, 28181168, 63045684, 141071758, 315668674, 706452161, 1581088178, 3538954508, 7921759060, 17733983146, 39702719910, 88893039358
Offset: 9

Views

Author

Alois P. Heinz, Jul 04 2014

Keywords

Comments

In a rooted tree with thinning limbs the outdegree of a parent node is larger than or equal to the outdegree of any of its child nodes.

Links

Crossrefs

Column k=8 of A244657.

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n b(n-1$2, 8$2):
seq(a(n), n=9..50);
Showing 1-10 of 13 results. Next

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