A317708 -id:A317708 - cp's OEIS frontend

A306200 Number of unlabeled rooted semi-identity trees with n nodes.

0, 1, 1, 2, 4, 8, 18, 41, 98, 237, 591, 1488, 3805, 9820, 25593, 67184, 177604, 472177, 1261998, 3388434, 9136019, 24724904, 67141940, 182892368, 499608724, 1368340326, 3756651116, 10336434585, 28499309291, 78727891420, 217870037932, 603934911859, 1676720329410

Offset: 0

Gus Wiseman, Jan 29 2019

nonn

A rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees.

			The a(1) = 1 through a(7) = 8 trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (o(o))   (o(oo))    (o(ooo))
                 (((o)))  (oo(o))    (oo(oo))
                          (((oo)))   (ooo(o))
                          ((o(o)))   (((ooo)))
                          (o((o)))   ((o)(oo))
                          ((((o))))  ((o(oo)))
                                     ((oo(o)))
                                     (o((oo)))
                                     (o(o(o)))
                                     (oo((o)))
                                     ((((oo))))
                                     (((o(o))))
                                     ((o)((o)))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

Alois P. Heinz, Table of n, a(n) for n = 0..2166

Cf. A000081, A004111, A276625, A301700, A306201, A316471, A316474, A317708, A317712, A317718.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      add(b(n-i*j, i-1)*binomial(a(i), j), j=0..n/i))
    end:
a:= n-> `if`(n=0, 0, b(n-1$2)):
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 29 2019

ursit[n_]:=Join@@Table[Select[Union[Sort/@Tuples[ursit/@ptn]],UnsameQ@@DeleteCases[#,{}]&],{ptn,IntegerPartitions[n-1]}];
Table[Length[ursit[n]],{n,10}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1,
     Sum[b[n - i*j, i - 1]*Binomial[a[i], j], {j, 0, n/i}]];
a[n_] := If[n == 0, 0, b[n - 1, n - 1]];
a /@ Range[0, 35] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

More terms from Alois P. Heinz, Jan 29 2019

A317712 Number of uniform rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 35, 72, 169, 388, 934, 2234, 5508, 13557, 33883, 85017, 215091, 546496, 1396524, 3582383, 9228470, 23852918, 61857180, 160871716, 419516462, 1096671326, 2873403980, 7544428973, 19847520789, 52308750878, 138095728065, 365153263313, 966978876376

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

An unlabeled rooted tree is uniform if the multiplicities of the branches directly under any given node are all equal.

			The a(5) = 8 uniform rooted trees:
  ((((o))))
  (((oo)))
  ((o(o)))
  ((ooo))
  (o((o)))
  (o(oo))
  ((o)(o))
  (oooo)

Vaclav Kotesovec, Table of n, a(n) for n = 1..2250 (terms 1..200 from Andrew Howroyd)
Gus Wiseman, All 72 uniform rooted trees with 8 nodes.

Cf. A000081, A001190, A004111, A072774, A301700, A317588.

Cf. A317705, A317707, A317708, A317709, A317710, A317711, A317717, A317718.

purt[n_]:=Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]],SameQ@@Length/@Split[#]&],{ptn,IntegerPartitions[n-1]}];
Table[Length[purt[n]],{n,10}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=[1]); for(n=2, n, my(t=WeighT(v)); v=concat(v, sumdiv(n-1, d, t[d]))); v} \\ Andrew Howroyd, Aug 28 2018

a(n) ~ c * d^n / n^(3/2), where d = 2.774067238136373782458114960391469140405537808253... and c = 0.43338208953061974806801546569720246018271214... - Vaclav Kotesovec, Sep 07 2019

Term a(21) and beyond from Andrew Howroyd, Aug 28 2018

A317710 Uniform tree numbers. Matula-Goebel numbers of uniform rooted trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 62, 64, 65, 66, 67, 69, 70, 73, 77, 78, 79, 81, 82, 83, 85, 86, 87, 91, 93, 94, 95, 97

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

A positive integer n is a uniform tree number iff either n = 1 or n is a power of a squarefree number whose prime indices are also uniform tree numbers. A prime index of n is a number m such that prime(m) divides n.

A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

Cf. A061775, A072774, A111299, A214577, A276625, A277098, A303431, A317589.

Cf. A317705, A317707, A317708, A317709, A317711 (complement), A317712, A317717, A317718.

rupQ[n_]:=Or[n==1,And[SameQ@@FactorInteger[n][[All,2]],And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]];
Select[Range[100],rupQ]

A317707 Number of powerful rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 11, 13, 22, 29, 46, 57, 94, 115, 180, 230, 349, 435, 671, 830, 1245, 1572, 2320, 2894, 4287, 5328, 7773, 9752, 14066, 17547, 25328, 31515, 45010, 56289, 79805, 99467, 140778, 175215, 246278, 307273, 429421, 534774, 745776, 927776, 1287038

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

An unlabeled rooted tree is powerful if either it is a single node or a single node with a single powerful tree as a branch, or if the branches of the root all appear with multiplicities greater than 1 and are themselves powerful trees.

			The a(7) = 11 powerful rooted trees:
  ((((((o))))))
  (((((oo)))))
  ((((ooo))))
  ((((o)(o))))
  (((oooo)))
  ((ooooo))
  (((o))((o)))
  ((oo)(oo))
  ((o)(o)(o))
  (oo(o)(o))
  (oooooo)

Alois P. Heinz, Table of n, a(n) for n = 1..8000

Cf. A000081, A001190, A001694, A004111, A301700, A303431, A317102.

Cf. A317705, A317708, A317709, A317710, A317711, A317712, A317718, A317719.

h:= proc(n, k, t) option remember; `if`(k=0, binomial(n+t, t),
      `if`(n=0, 0, add(h(n-1, k-j, t+1), j=2..k)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*h(a(i), j, 0), j=0..n/i)))
    end:
a:= proc(n) option remember; `if`(n<2, n, b(n-1$2)+a(n-1)) end:
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 31 2018

purt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]],Or[Length[#]==1,Min@@Length/@Split[#]>1]&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[purt[n]],{n,10}]
(* Second program: *)
h[n_, k_, t_] := h[n, k, t] = If[k == 0, Binomial[n + t, t], If[n == 0, 0, Sum[h[n - 1, k - j, t + 1], {j, 2, k}]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]* h[a[i], j, 0], {j, 0, n/i}]]];
a[n_] := a[n] = If[n < 2, n, b[n - 1, n - 1] + a[n - 1]];
Array[a, 50] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

a(27)-a(45) from Alois P. Heinz, Aug 31 2018

A317705 Matula-Goebel numbers of series-reduced powerful rooted trees.

Original entry on oeis.org

1, 4, 8, 16, 32, 49, 64, 128, 196, 256, 343, 361, 392, 512, 784, 1024, 1372, 1444, 1568, 2048, 2401, 2744, 2809, 2888, 3136, 4096, 5488, 5776, 6272, 6859, 8192, 9604, 10976, 11236, 11552, 12544, 16384, 16807, 17161, 17689, 19208, 21952, 22472, 23104, 25088

Offset: 1

Gus Wiseman, Aug 04 2018

nonn

A positive integer n is a Matula-Goebel number of a series-reduced powerful rooted tree iff either n = 1 or n is a powerful number (meaning its prime multiplicities are all greater than 1) whose prime indices are all Matula-Goebel numbers of series-reduced powerful rooted trees, where a prime index of n is a number m such that prime(m) divides n.

			The sequence of Matula-Goebel numbers of series-reduced powerful rooted trees together with the corresponding trees begins:
    1: o
    4: (oo)
    8: (ooo)
   16: (oooo)
   32: (ooooo)
   49: ((oo)(oo))
   64: (oooooo)
  128: (ooooooo)
  196: (oo(oo)(oo))
  256: (oooooooo)
  343: ((oo)(oo)(oo))
  361: ((ooo)(ooo))
  392: (ooo(oo)(oo))
  512: (ooooooooo)
  784: (oooo(oo)(oo))

Index entries for sequences related to Matula-Göbel numbers

Cf. A000081, A001694, A061775, A111299, A214577, A276625, A277098, A303431.

Cf. A317102, A317707, A317708, A317709, A317710, A317711, A317712, A317717, A317718, A317719.

powgoQ[n_]:=Or[n==1,And[Min@@FactorInteger[n][[All,2]]>1,And@@powgoQ/@PrimePi/@FactorInteger[n][[All,1]]]];
Select[Range[1000],powgoQ] (* Gus Wiseman, Aug 31 2018 *)
(* Second program: *)
Nest[Function[a, Append[a, Block[{k = a[[-1]] + 1}, While[Nand[AllTrue[#[[All, -1]], # > 1 & ], AllTrue[PrimePi[#[[All, 1]] ], MemberQ[a, #] &]] &@ FactorInteger@ k, k++]; k]]], {1}, 44] (* Michael De Vlieger, Aug 05 2018 *)

Rewritten by Gus Wiseman, Aug 31 2018

A317718 Number of uniform relatively prime rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 27, 55, 125, 278, 650, 1510, 3624, 8655, 21017, 51212, 125857, 310581, 770767, 1920226

Offset: 1

Gus Wiseman, Aug 05 2018

An unlabeled rooted tree is uniform and relatively prime iff either it is a single node or a single node with a single uniform relatively prime branch, or the branches of the root have empty intersection (relatively prime) and equal multiplicities (uniform) and are themselves uniform relatively prime trees.

			The a(6) = 13 uniform relatively prime rooted trees:
  (((((o)))))
  ((((oo))))
  (((o(o))))
  (((ooo)))
  ((o((o))))
  ((o(oo)))
  ((oooo))
  (o(((o))))
  (o((oo)))
  (o(o(o)))
  (o(ooo))
  ((o)((o)))
  (ooooo)

A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.

Cf. A000081, A001190, A004111, A072774, A301700, A317588.

Cf. A317705, A317707, A317708, A317709, A317710, A317711, A317712, A317717.

purt[n_]:=purt[n]=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]],Or[Length[#]==1,And[SameQ@@Length/@Split[#],Intersection@@#=={}]]&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[purt[n]],{n,20}]

A317709 Aperiodic relatively prime tree numbers. Matula-Goebel numbers of aperiodic relatively prime trees.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 12, 13, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 37, 40, 41, 44, 45, 47, 48, 50, 52, 54, 55, 58, 60, 61, 62, 66, 71, 72, 74, 75, 78, 79, 80, 82, 88, 89, 90, 93, 94, 96, 99, 101, 104, 108, 109, 110, 113, 116, 120, 122, 123, 124, 127, 130

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

A positive integer n is in the sequence iff either n = 1 or n is a prime number whose prime index already belongs to the sequence or n is not a perfect power and its prime indices are relatively prime numbers already belonging to the sequence. A prime index of n is a number m such that prime(m) divides n.

			The sequence of aperiodic relatively prime tree numbers together with their Matula-Goebel trees begins:
   1: o
   2: (o)
   3: ((o))
   5: (((o)))
   6: (o(o))
  10: (o((o)))
  11: ((((o))))
  12: (oo(o))
  13: ((o(o)))
  15: ((o)((o)))
  18: (o(o)(o))
  20: (oo((o)))
  22: (o(((o))))
  24: (ooo(o))
  26: (o(o(o)))
  29: ((o((o))))
  30: (o(o)((o)))
  31: (((((o)))))

Cf. A000081, A061775, A111299, A214577, A276625, A277098, A301700, A303431, A316503.

Cf. A317705, A317707, A317708, A317710, A317711, A317712.

rupQ[n_]:=Or[n==1,If[PrimeQ[n],rupQ[PrimePi[n]],And[GCD@@FactorInteger[n][[All,2]]==1,GCD@@PrimePi/@FactorInteger[n][[All,1]]==1,And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[100],rupQ]

A317717 Uniform relatively prime tree numbers. Matula-Goebel numbers of uniform relatively prime rooted trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 22, 26, 29, 30, 31, 32, 33, 34, 35, 36, 38, 41, 42, 43, 47, 51, 53, 55, 58, 59, 62, 64, 66, 67, 70, 77, 78, 79, 82, 85, 86, 93, 94, 95, 100, 101, 102, 105, 106, 109, 110, 113, 114, 118, 119, 123, 127, 128

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

A positive integer n is a uniform relatively prime tree number iff either n = 1 or n is a prime number whose prime index is a uniform relatively prime tree number, or n is a power of a squarefree number whose prime indices are relatively prime and are themselves uniform relatively prime tree numbers. A prime index of n is a number m such that prime(m) divides n.

A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

Cf. A000081, A061775, A072774, A214577, A276625, A277098, A317588.

Cf. A317705, A317707, A317708, A317709, A317710, A317711, A317712, A317718.

rupQ[n_]:=Or[n==1,If[PrimeQ[n],rupQ[PrimePi[n]],And[SameQ@@FactorInteger[n][[All,2]],GCD@@PrimePi/@FactorInteger[n][[All,1]]==1,And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[200],rupQ]

A317711 Numbers that are not uniform tree numbers.

Original entry on oeis.org

12, 18, 20, 24, 28, 37, 40, 44, 45, 48, 50, 52, 54, 56, 60, 61, 63, 68, 71, 72, 74, 75, 76, 80, 84, 88, 89, 90, 92, 96, 98, 99, 104, 107, 108, 111, 112, 116, 117, 120, 122, 124, 126, 132, 135, 136, 140, 142, 144, 147, 148, 150, 152, 153, 156, 157, 160, 162

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

A positive integer n is a uniform tree number iff either n = 1 or n is a power of a squarefree number whose prime indices are also uniform tree numbers. A prime index of n is a number m such that prime(m) divides n.

			The sequence of non-uniform tree numbers together with their Matula-Goebel trees begins:
  12: (oo(o))
  18: (o(o)(o))
  20: (oo((o)))
  24: (ooo(o))
  28: (oo(oo))
  37: ((oo(o)))
  40: (ooo((o)))
  44: (oo(((o))))
  45: ((o)(o)((o)))
  48: (oooo(o))
  50: (o((o))((o)))
  52: (oo(o(o)))
  54: (o(o)(o)(o))
  56: (ooo(oo))
  60: (oo(o)((o)))

Cf. A061775, A072774, A111299, A214577, A276625, A277098, A303431, A317590.

Cf. A317705, A317707, A317708, A317709, A317710 (complement), A317712, A317717, A317718.

rupQ[n_]:=Or[n==1,And[SameQ@@FactorInteger[n][[All,2]],And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]];
Select[Range[100],!rupQ[#]&]

A317787 Number of locally nonintersecting rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 40, 95, 227, 557, 1382, 3485, 8865, 22790, 59022, 153972, 404066, 1066236, 2826885, 7527411, 20121154

Offset: 1

Gus Wiseman, Aug 07 2018

An unlabeled rooted tree is locally nonintersecting if there is no common subbranch to all branches directly under any given node.

			The a(6) = 18 locally nonintersecting rooted trees:
  (((((o)))))
  ((((oo))))
  (((o(o))))
  ((o((o))))
  (o(((o))))
  ((o)((o)))
  (((ooo)))
  ((o(oo)))
  ((oo(o)))
  (o((oo)))
  (o(o(o)))
  (oo((o)))
  (o(o)(o))
  ((oooo))
  (o(ooo))
  (oo(oo))
  (ooo(o))
  (ooooo)
Missing from this list are (((o)(o))) and ((o)(oo)).

Cf. A000081, A276625, A301700, A316473, A316475, A316501, A316502, A317708, A317785, A317789.

rurt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]],Or[Length[#]==1,Intersection@@#=={}]&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[rurt[n]],{n,10}]

a(16)-a(21) from Robert Price, Sep 16 2018

cp's OEIS Frontend

A306200 Number of unlabeled rooted semi-identity trees with n nodes.

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A317712 Number of uniform rooted trees with n nodes.

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A317710 Uniform tree numbers. Matula-Goebel numbers of uniform rooted trees.

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A317707 Number of powerful rooted trees with n nodes.

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A317705 Matula-Goebel numbers of series-reduced powerful rooted trees.

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A317718 Number of uniform relatively prime rooted trees with n nodes.

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A317709 Aperiodic relatively prime tree numbers. Matula-Goebel numbers of aperiodic relatively prime trees.

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Mathematica

A317717 Uniform relatively prime tree numbers. Matula-Goebel numbers of uniform relatively prime rooted trees.

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