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 1-5 of 5 results.

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

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[#]]&]

A298478 Number of unlabeled rooted trees with n nodes in which all positive outdegrees are different.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 13, 15, 23, 34, 95, 106, 176, 241, 374, 942, 1129, 1760, 2515, 3711, 5136, 12857, 14911, 23814, 33002, 49141, 65798, 97056, 209707, 255042, 389725, 545290, 790344, 1071010, 1525919, 2043953, 4272124, 5110583, 7772247, 10611491, 15447864, 20496809
Offset: 1

Views

Author

Gus Wiseman, Jan 19 2018

Keywords

Comments

a(n) is the number of labeled trees with sum of the labels equal to n-1 and the outdegree of every node less than or equal to the value of its label. - Andrew Howroyd, Feb 02 2021

Examples

			The a(7) = 13 trees: ((o(ooo))), ((oo(oo))), ((ooooo)), (o((ooo))), (o(oo(o))), (o(oooo)), ((o)(ooo)), (oo((oo))), (oo(o(o))), (o(o)(oo)), (ooo(oo)), (oooo(o)), (oooooo).

Links

Crossrefs

Cf. A000081, A001190, A001678, A004111, A032305, A124343, A290689, A295461, A298118, A298304, A298422, A298479.

Programs

  • Mathematica
    krut[n_]:=krut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[krut/@c]]]/@IntegerPartitions[n-1],UnsameQ@@Length/@Cases[#,{},{0,Infinity}]&]];
Table[krut[n]//Length,{n,15}]
  • PARI
    relabel(b)={my(w=hammingweight(b)); b = bitand((1<Andrew Howroyd, Feb 02 2021

Extensions

a(27)-a(34) from Robert G. Wilson v, Jan 19 2018
Terms a(35) and beyond from Andrew Howroyd, Feb 02 2021

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 1-5 of 5 results.

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