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 1-4 of 4 results.

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 *)

A298305 Matula-Goebel numbers of rooted trees with strictly thinning limbs.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 16, 18, 24, 27, 28, 32, 36, 42, 48, 52, 54, 56, 63, 64, 72, 78, 81, 84, 92, 96, 98, 104, 108, 112, 117, 126, 128, 138, 144, 147, 152, 156, 162, 168, 182, 184, 189, 192, 196, 207, 208, 216, 224, 228, 234, 243, 252, 256, 273, 276, 288, 294
Offset: 1

Views

Author

Gus Wiseman, Jan 16 2018

Keywords

Comments

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

Examples

			Sequence of trees begins:
1  o
2  (o)
4  (oo)
6  (o(o))
8  (ooo)
9  ((o)(o))
12 (oo(o))
16 (oooo)
18 (o(o)(o))
24 (ooo(o))
27 ((o)(o)(o))
28 (oo(oo))
32 (ooooo)
36 (oo(o)(o))
42 (o(o)(oo))
48 (oooo(o))
52 (oo(o(o)))
54 (o(o)(o)(o))
56 (ooo(oo))
63 ((o)(o)(oo))
64 (oooooo)
72 (ooo(o)(o))
78 (o(o)(o(o)))
81 ((o)(o)(o)(o))
84 (oo(o)(oo))
92 (oo((o)(o)))
96 (ooooo(o))
98 (o(oo)(oo))

Crossrefs

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

Programs

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

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

Views

Author

Gus Wiseman, Jan 16 2018

Keywords

Comments

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

Crossrefs

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

Programs

  • Mathematica
    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[#]]&]

A298363 Matula-Goebel numbers of rooted identity trees with thinning limbs.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 15, 22, 26, 30, 31, 33, 39, 55, 58, 62, 65, 66, 78, 87, 93, 94, 110, 127, 130, 141, 143, 145, 155, 158, 165, 174, 186, 195, 202, 235, 237, 254, 274, 282, 286, 290, 303, 310, 319, 330, 334, 341, 377, 381, 390, 395, 403, 411, 429, 435, 465
Offset: 1

Views

Author

Gus Wiseman, Jan 17 2018

Keywords

Comments

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

Examples

			Sequence of trees begins:
1  o
2  (o)
3  ((o))
5  (((o)))
6  (o(o))
10 (o((o)))
11 ((((o))))
15 ((o)((o)))
22 (o(((o))))
26 (o(o(o)))
30 (o(o)((o)))
31 (((((o)))))
33 ((o)(((o))))
39 ((o)(o(o)))
55 (((o))(((o))))
58 (o(o((o))))
62 (o((((o)))))
65 (((o))(o(o)))
66 (o(o)(((o))))
78 (o(o)(o(o)))
87 ((o)(o((o))))
93 ((o)((((o)))))
94 (o((o)((o))))

Crossrefs

Cf. A000081, A004111, A007097, A061775, A111299, A124343, A124346, A214577, A276625, A277098, A290689, A290760, A298304, A298305.

Programs

  • Mathematica
    MGtree[n_]:=If[n===1,{},MGtree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
idthinQ[t_]:=And@@Cases[t,b_List:>UnsameQ@@b&&Length[b]>=Max@@Length/@b,{0,Infinity}];
Select[Range[500],idthinQ[MGtree[#]]&]

Formula

Intersection of A276625 and A298303.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.