A317718 -id:A317718 - 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

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[#]&]

A319269 Number of uniform factorizations of n into factors > 1, where a factorization is uniform if all factors appear with the same multiplicity.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 5, 1, 4, 2, 2, 2, 7, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 8, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 4, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Sep 16 2018

nonn

			The a(144) = 17 factorizations:
  (144),
  (2*72), (3*48), (4*36), (6*24), (8*18), (9*16), (12*12),
  (2*3*24), (2*4*18), (2*6*12), (2*8*9), (3*4*12), (3*6*8),
  (2*2*6*6), (2*3*4*6), (3*3*4*4).

Cf. A001055, A045778, A047966, A052409, A072774, A296132, A303386, A317710, A317712, A317717, A317718.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],SameQ@@Length/@Split[#]&]],{n,100}]

a(n) = Sum_{d|A052409(n)} A045778(n^(1/d)).

A317719 Numbers that are not powerful tree numbers.

Original entry on oeis.org

6, 10, 12, 13, 14, 15, 18, 20, 21, 22, 24, 26, 28, 29, 30, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82, 84, 85, 86, 87, 88, 89, 90, 91

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

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

			The sequence of numbers that are not powerful tree numbers together with their Matula-Goebel trees begins:
   6: (o(o))
  10: (o((o)))
  12: (oo(o))
  13: ((o(o)))
  14: (o(oo))
  15: ((o)((o)))
  18: (o(o)(o))
  20: (oo((o)))
  21: ((o)(oo))
  22: (o(((o))))
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  29: ((o((o))))
  30: (o(o)((o)))

Complement of A318612.
Cf. A000081, A001694, A061775, A111299, A214577, A276625, A277098, A303431.
Cf. A317102, A317707, A317708, A317709, A317710, A317711, A317712, A317717, A317718.

powgoQ[n_]:=Or[n==1,If[PrimeQ[n],powgoQ[PrimePi[n]],And[Min@@FactorInteger[n][[All,2]]>1,And@@powgoQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[100],!powgoQ[#]&]

A318689 Number of powerful uniform rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 11, 12, 19, 23, 35, 36, 63, 64, 98, 112, 173, 174, 291, 292, 473, 509, 791, 792, 1345, 1356, 2158, 2257, 3634, 3635, 6053, 6054, 9807, 10091, 16173, 16216, 26783, 26784, 43076, 43880, 70631, 70632, 114975, 114976, 184665, 186996, 298644, 298645, 481978, 482011

Offset: 1

Gus Wiseman, Aug 31 2018

nonn

A powerful uniform rooted tree with n nodes is either a single powerful uniform branch with n-1 nodes, or a powerful uniform multiset (all multiplicities are equal to the same number > 1) of powerful uniform rooted trees with a total of n-1 nodes.

			The a(8) = 12 powerful uniform rooted trees:
  (((((((o)))))))
  ((((((oo))))))
  (((((o)(o)))))
  ((((o))((o))))
  (((((ooo)))))
  (((o)(o)(o)))
  ((((oooo))))
  (((oo)(oo)))
  ((oo(o)(o)))
  (((ooooo)))
  ((oooooo))
  (ooooooo)

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Gus Wiseman, All 35 powerful uniform rooted trees with 11 nodes.

Cf. A000081, A003238, A072774, A317705, A317707, A317710, A317717, A317718, A318611, A318612, A318690, A318691, A318692.

rurt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]],Or[Length[#]==1,And[Min@@Length/@Split[#]>=2,SameQ@@Length/@Split[#]]]&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[rurt[n]],{n,15}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=vector(n)); v[1]=1; for(n=1, n-1, my(u=WeighT(v[1..n])); v[n+1] = sumdiv(n,d,u[d]) - u[n] + v[n]); v} \\ Andrew Howroyd, Dec 09 2020

Terms a(21) and beyond from Andrew Howroyd, Dec 09 2020

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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Maple

Mathematica

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

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Mathematica

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

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A317711 Numbers that are not uniform tree numbers.

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A319269 Number of uniform factorizations of n into factors > 1, where a factorization is uniform if all factors appear with the same multiplicity.

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