A004111 -id:A004111 - cp's OEIS frontend

A301462 Number of enriched r-trees of size n.

1, 2, 3, 8, 23, 77, 254, 921, 3249, 12133, 44937, 172329, 654895, 2565963, 9956885, 39536964, 156047622, 626262315, 2499486155, 10129445626, 40810378668, 166475139700, 676304156461, 2775117950448, 11342074888693, 46785595997544, 192244951610575, 796245213910406

Offset: 0

Gus Wiseman, Mar 21 2018

nonn

An enriched r-tree of size n > 0 is either a single node of size n, or a finite sequence of enriched r-trees with weakly decreasing sizes summing to n - 1.

These are different from the R-trees of data science and the enriched R-trees of Bousquet-Mélou and Courtiel.

			The a(3) = 8 enriched r-trees: 3, (2), ((1)), ((())), (11), (1()), (()1), (()()).

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A000081, A003238, A004111, A032305, A055277, A093637, A127524, A196545, A289501, A290689, A300443, A301342-A301345, A301364-A301368, A301422, A301467, A301469, A301470.

ert[n_]:=ert[n]=1+Sum[Times@@ert/@y,{y,IntegerPartitions[n-1]}];
Array[ert,30]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x^n)), n-1)); concat([1], v)} \\ Andrew Howroyd, Aug 26 2018

O.g.f.: 1/(1 - x) + x Product_{i > 0} 1/(1 - a(i) x^i).

A279861 Number of transitive finitary sets with n brackets. Number of transitive rooted identity trees with n nodes.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 2, 2, 2, 5, 4, 6, 8, 10, 14, 23, 26, 34, 46, 64, 81, 115, 158, 199, 277, 376, 505, 684, 934, 1241, 1711, 2300, 3123, 4236, 5763, 7814, 10647, 14456, 19662

Offset: 1

Gus Wiseman, Dec 21 2016

nonn

A finitary set is transitive if every element is also a subset. Transitive sets are also called full sets.

			Sequence of transitive finitary sets begins:
1  ()
2  (())
4  (()(()))
7  (()(())((())))
8  (()(())(()(())))
11 (()(())((()))(((()))))
   (()(())((()))(()(())))
12 (()(())((()))(()((()))))
13 (()(())((()))((())((()))))
   (()(())(()(()))((()(()))))
14 (()(())((()))(()(())((()))))
   (()(())(()(()))(()(()(()))))
15 (()(())((()))(((())))(()(())))
   (()(())(()(()))((())(()(()))))
16 (()(())((()))(((())))((((())))))
   (()(())((()))(((())))(()((()))))
   (()(())((()))(()(()))(()((()))))
   (()(())((()))(()(()))((()(()))))
   (()(())(()(()))(()(())(()(()))))
17 (()(())((()))(((())))(()(((())))))
   (()(())((()))(((())))((())((()))))
   (()(())((()))(()(()))(()(()(()))))
   (()(())((()))(()(()))((())((()))))
18 (()(())((()))(((())))((())(((())))))
   (()(())((()))(((())))(()(())((()))))
   (()(())((()))(()(()))((())(()(()))))
   (()(())((()))(()(()))(()(())((()))))
   (()(())((()))((()((()))))(()((()))))
   (()(())((()))(()((())))((())((()))))

Wikipedia, Transitive set
Gus Wiseman, Transitive rooted identity trees example n=23

Cf. A001192, A004111, A061773, A279614, A276625, A279065, A279863.

transfins[n_]:=transfins[n]=If[n===1,{{}},Select[Union@@FixedPointList[Complement[Union@@Function[fin,Cases[Complement[Subsets[fin],fin],sub_:>With[{nov=Sort[Append[fin,sub]]},nov/;Count[nov,_List,{0,Infinity}]<=n]]]/@#,#]&,Array[transfins,n-1,1,Union]],Count[#,_List,{0,Infinity}]===n&]];
Table[Length[transfins[n]],{n,20}]

A301467 Number of enriched r-trees of size n with no empty subtrees.

Original entry on oeis.org

1, 2, 4, 8, 20, 48, 136, 360, 1040, 2944, 8704, 25280, 76320, 226720, 692992, 2096640, 6470016, 19799936, 61713152, 190683520, 598033152, 1863995392, 5879859200, 18438913536, 58464724992, 184356152832, 586898946048, 1859875518464, 5941384080384, 18901502482432

Offset: 1

Gus Wiseman, Mar 21 2018

nonn

An enriched r-tree of size n > 0 with no empty subtrees is either a single node of size n, or a finite nonempty sequence of enriched r-trees with no empty subtrees and with weakly decreasing sizes summing to n - 1.

			The a(4) = 8 enriched r-trees with no empty subtrees: 4, (3), (21), ((2)), (111), ((11)), ((1)1), (((1))).
The a(5) = 20 enriched r-trees with no empty subtrees:
  5,
  (4), ((3)), ((21)), (((2))), ((111)), (((11))), (((1)1)), ((((1)))),
  (31), (22), (2(1)), ((2)1), ((1)2), ((11)1), ((1)(1)), (((1))1),
  (211), ((1)11),
  (1111).

Alois P. Heinz, Table of n, a(n) for n = 1..1910

Cf. A000081, A004111, A032305, A055277, A093637, A127524, A196545, A289501, A300660, A301342-A301345, A301364-A301368, A301422, A301462, A301469, A301470.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)* a(i)^j, j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, 1+b(n-1$2)):
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 21 2018

pert[n_]:=pert[n]=If[n===1,1,1+Sum[Times@@pert/@y,{y,IntegerPartitions[n-1]}]];
Array[pert,30]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
     Sum[b[n - i*j, i - 1] a[i]^j, {j, 0, n/i}]]];
a[n_] := a[n] = If[n < 2, n, 1 + b[n-1, n-1]];
Array[a, 30] (* Jean-François Alcover, May 09 2021, after Alois P. Heinz *)

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, v[n] = 1 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x^n)), n-1)); v} \\ Andrew Howroyd, Aug 26 2018

O.g.f.: x^2/(1 - x) + x Product_{i > 0} 1/(1 - a(i) x^i).

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

Original entry on oeis.org

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

A318185 Number of totally transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 12, 17, 28, 41, 65, 96, 150, 221, 342, 506, 771, 1142, 1731, 2561, 3855, 5702, 8538, 12620, 18817, 27774, 41276, 60850, 90139

Offset: 1

Gus Wiseman, Aug 20 2018

A rooted tree is totally transitive if every branch of the root is totally transitive and every branch of a branch of the root is also a branch of the root. Unlike transitive rooted trees (A290689), every terminal subtree of a totally transitive rooted tree is itself totally transitive.

			The a(8) = 12 totally transitive rooted trees:
  (o(o)(o(o)))
  (o(o)(o)(o))
  (o(o)(ooo))
  (o(oo)(oo))
  (oo(o)(oo))
  (ooo(o)(o))
  (o(ooooo))
  (oo(oooo))
  (ooo(ooo))
  (oooo(oo))
  (ooooo(o))
  (ooooooo)
The a(9) = 17 totally transitive rooted trees:
  (o(o)(oo(o)))
  (oo(o)(o(o)))
  (o(o)(o)(oo))
  (oo(o)(o)(o))
  (o(o)(oooo))
  (o(oo)(ooo))
  (oo(o)(ooo))
  (oo(oo)(oo))
  (ooo(o)(oo))
  (oooo(o)(o))
  (o(oooooo))
  (oo(ooooo))
  (ooo(oooo))
  (oooo(ooo))
  (ooooo(oo))
  (oooooo(o))
  (oooooooo)

Cf. A000081, A001678, A004111, A279861, A290689, A290760, A290822, A318186, A318187.

totra[n_]:=totra[n]=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[totra/@c]],Complement[Union@@#,#]=={}&],{c,IntegerPartitions[n-1]}]];
Table[Length[totra[n]],{n,20}]

A301422 Regular triangle where T(n,k) is the number of r-trees of size n with k leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 6, 8, 4, 1, 0, 1, 9, 19, 14, 5, 1, 0, 1, 12, 36, 40, 21, 6, 1, 0, 1, 16, 65, 102, 75, 30, 7, 1, 0, 1, 20, 106, 223, 224, 123, 40, 8, 1, 0, 1, 25, 168, 457, 604, 439, 191, 52, 9, 1, 0, 1, 30, 248, 847, 1433, 1346, 764, 276

Offset: 1

Gus Wiseman, Mar 20 2018

An r-tree (A093637) of size n > 0 is a finite sequence of r-trees with weakly decreasing sizes summing to n - 1. This is a similar construction to p-trees (A196545) except that r-trees are not required to be series-reduced and are weighted by all nodes (including the root) rather than just the leaves.

			Triangle begins:
  1
  1   0
  1   1   0
  1   2   1   0
  1   4   3   1   0
  1   6   8   4   1   0
  1   9  19  14   5   1   0
  1  12  36  40  21   6   1   0
  1  16  65 102  75  30   7   1   0
  1  20 106 223 224 123  40   8   1   0
  1  25 168 457 604 439 191  52   9   1   0
  ...
The T(6,3) = 8 r-trees: (((ooo))), (((oo)o)), (((o)oo)), (((oo))o), (((o)o)o), ((oo)(o)), (((o))oo), ((o)(o)o).

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Cf. A000081, A003238, A004111, A032305, A055277, A093637, A127524, A196545, A289501, A290689, A291443, A297791, A300443, A301342-A301345, A301364.

rtrees[n_]:=Join@@Table[Tuples[rtrees/@y],{y,IntegerPartitions[n-1]}];
Table[Length[Select[rtrees[n],Count[#,{},{-2}]===k&]],{n,8},{k,n}]

A(n)={my(v=vector(n)); v[1]=y; for(n=2, n, v[n] = polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x^n)), n-1)); vector(n, k, Vecrev(v[k]/y,k))}
{ my(T=A(10)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Aug 26 2018

A316471 Number of locally disjoint rooted identity trees with n nodes, meaning no branch overlaps any other branch of the same root.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 11, 21, 43, 89, 185, 391, 840, 1822, 3975, 8727, 19280, 42841, 95661, 214490

Offset: 1

Gus Wiseman, Jul 04 2018

			The a(7) = 11 locally disjoint rooted identity trees:
((((((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)))

Gus Wiseman, The a(8) = 21 locally disjoint rooted identity trees.

Cf. A000081, A004111, A276625, A277098, A302696, A303362, A304713, A316467, A316471, A316473, A316474, A316494.

strut[n_]:=strut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@IntegerPartitions[n-1],UnsameQ@@#&&Select[Tuples[#,2],UnsameQ@@#&&(Intersection@@#=!={})&]=={}&]];
Table[Length[strut[n]],{n,20}]

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

A324764 Number of anti-transitive rooted identity trees with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 9, 20, 41, 89, 196, 443, 987, 2246, 5114, 11757, 27122, 62898, 146392, 342204, 802429, 1887882

Offset: 1

Gus Wiseman, Mar 17 2019

A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. It is anti-transitive if the branches of the branches of the root are disjoint from the branches of the root.
Also the number of finitary sets S with n brackets where no element of an element of S is also an element of S. For example, the a(8) = 20 finitary sets 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}}}}}
  {{o},{{o,{o}}}}
  {{o},{o,{{o}}}}
  {{{o}},{o,{o}}}

			The a(1) = 1 through a(7) = 9 anti-transitive rooted identity trees:
  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))))))

Gus Wiseman, The a(9) = 41 anti-transitive rooted identity trees.

Cf. A000081, A004111, A276625, A279861, A290689, A304360, A306844 (non-identity version), A316500, A317787, A318185.

Cf. A324694, A324751, A324756, A324758, A324765, A324767, A324768, A324770, A324839, A324840, A324844.

idall[n_]:=If[n==1,{{}},Select[Union[Sort/@Join@@(Tuples[idall/@#]&/@IntegerPartitions[n-1])],UnsameQ@@#&]];
Table[Length[Select[idall[n],Intersection[Union@@#,#]=={}&]],{n,10}]

a(21)-a(22) from Jinyuan Wang, Jun 20 2020

A045648 Number of chiral n-ominoes in (n-1)-space, one cell labeled.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 34, 75, 166, 370, 841, 1937, 4488, 10470, 24617, 58237, 138435, 330563, 792745, 1908379, 4609434, 11167781, 27134824, 66102921, 161417867, 395042562, 968791315, 2380383481, 5859176855, 14446043494, 35672895787, 88219204394, 218466647493

Offset: 1

Robert A. Russell

Needed for generating chiral n-ominoes in (n-1)-space with no cells labeled, Lunnon's DR(n, n-1) - DE(n, n-1). Knuth describes a method for a similar enumeration, that of free trees with n nodes.

Euler transform of a(n) - if(n%4!=2, 0, a(n/2)) is sequence itself with offset 0.

D. E. Knuth, Fundamental Algorithms, 3d Ed. 1997, pp. 386-388.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
W. F. Lunnon, Counting Multidimensional Polyominoes, Computer Journal, Vol. 18 (1975), pp. 366-367.

Cf. A045649, A000081, A004111.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*(a(d)-
      `if`(irem(d, 4)=2, a(d/2), 0)), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> b(n-1):
seq(a(n), n=1..40);  # Alois P. Heinz, Feb 24 2015

s[ n_, k_ ] := s[ n, k ]=c[ n+1-k ]+If[ n<2k, 0, s[ n-k, k ](-1)^k ]; c[ 1 ]=1; c[ n_ ] := c[ n ]=Sum[ c[ i ]s[ n-1, i ]i, {i, 1, n-1} ]/(n-1); Table[ c[ i ], {i, 1, 30} ]

{a(n)=local(A=x); if(n<1, 0, for(k=1, n-1, A/=(1-(-x)^k+x*O(x^n))^((-1)^k*polcoeff(A, k))); polcoeff(A, n))} /* Michael Somos, Dec 16 2002 */

G.f.: A(x) = x exp(A(x) + A(-x^2)/2 + A(x^3)/3 + A(-x^4)/4 + ...).
Also A(x) = Sum_{n >= 1} a(n)*x^n = x / Product_{n >= 1} (1-(-x)^n)^((-1)^n*a(n)).
G.f.: x*Product_{n>0} (1-x^(4n-2))^a(2n-1)/(1-x^n)^a(n).
a(n) ~ c * d^n / n^(3/2), where d = 2.58968405406171542574769690513208346256... and c = 0.386431095907583923297618874742... . - Vaclav Kotesovec, Feb 29 2016

cp's OEIS Frontend

A301462 Number of enriched r-trees of size n.

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A279861 Number of transitive finitary sets with n brackets. Number of transitive rooted identity trees with n nodes.

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A301467 Number of enriched r-trees of size n with no empty subtrees.

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

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A318185 Number of totally transitive rooted trees with n nodes.

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A301422 Regular triangle where T(n,k) is the number of r-trees of size n with k leaves.

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A316471 Number of locally disjoint rooted identity trees with n nodes, meaning no branch overlaps any other branch of the same root.

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A317712 Number of uniform rooted trees with n nodes.

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