A291443 -id:A291443 - cp's OEIS frontend

A032305 Number of rooted trees where any 2 subtrees extending from the same node have a different number of nodes.

1, 1, 1, 2, 3, 6, 12, 25, 51, 111, 240, 533, 1181, 2671, 6014, 13795, 31480, 72905, 168361, 393077, 914784, 2150810, 5040953, 11914240, 28089793, 66702160, 158013093, 376777192, 896262811, 2144279852, 5120176632, 12286984432, 29428496034, 70815501209

Offset: 1

Christian G. Bower

nonn

			The a(6) = 6 fully unbalanced trees: (((((o))))), (((o(o)))), ((o((o)))), (o(((o)))), (o(o(o))), ((o)((o))). - _Gus Wiseman_, Jan 10 2018

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Gus Wiseman, Illustration of the first 8 terms of A032305.
Index entries for sequences related to rooted trees

Cf. A000081, A001678, A003238, A004111, A213920, A273873, A290689, A291443, A297571.

Column k=1 of A318753.

A:= proc(n) if n<=1 then x else convert(series(x* (product(1+ coeff(A(n-1), x,i)*x^i, i=1..n-1)), x=0, n+1), polynom) fi end: a:= n-> coeff(A(n), x,n): seq(a(n), n=1..31);  # Alois P. Heinz, Aug 22 2008
# second Maple program:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(j=0, 1, g((i-1)$2))*g(n-i*j, i-1), j=0..min(1, n/i))))
    end:
a:= n-> g((n-1)$2):
seq(a(n), n=1..35);  # Alois P. Heinz, Mar 04 2013

nn=30;f[x_]:=Sum[a[n]x^n,{n,0,nn}];sol=SolveAlways[0 == Series[f[x]-x Product[1+a[i]x^i,{i,1,nn}],{x,0,nn}],x];Table[a[n],{n,1,nn}]/.sol  (* Geoffrey Critzer, Nov 17 2012 *)
allnim[n_]:=If[n===1,{{}},Join@@Function[c,Select[Union[Sort/@Tuples[allnim/@c]],UnsameQ@@(Count[#,_List,{0,Infinity}]&/@#)&]]/@IntegerPartitions[n-1]];
Table[Length[allnim[n]],{n,15}] (* Gus Wiseman, Jan 10 2018 *)
g[n_, i_] := g[n, i] = If[n == 0, 1, If[i < 1, 0,
     Sum[If[j == 0, 1, g[i-1, i-1]]*g[n-i*j, i-1], {j, 0, Min[1, n/i]}]]];
a[n_] := g[n-1, n-1];
Array[a, 35] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

a(n)=polcoeff(x*prod(i=1,n-1,1+a(i)*x^i)+x*O(x^n),n)

Shifts left under "EFK" (unordered, size, unlabeled) transform.
G.f.: A(x) = x*Product_{n>=1} (1+a(n)*x^n) = Sum_{n>=1} a(n)*x^n. - Paul D. Hanna, Apr 07 2004
Lim_{n->infinity} a(n)^(1/n) = 2.5119824... - Vaclav Kotesovec, Nov 20 2019
G.f.: x * exp(Sum_{n>=1} Sum_{k>=1} (-1)^(k+1) * a(n)^k * x^(n*k) / k). - Ilya Gutkovskiy, Jun 30 2021

A298118 Number of unlabeled rooted trees with n nodes in which all positive outdegrees are odd.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 11, 21, 40, 80, 159, 322, 657, 1356, 2816, 5896, 12407, 26267, 55861, 119331, 255878, 550665, 1188786, 2574006, 5588177, 12162141, 26529873, 57993624, 127020653, 278716336, 612617523, 1348680531, 2973564157, 6565313455, 14514675376

Offset: 1

Gus Wiseman, Jan 12 2018

nonn

			The a(6) = 6 trees: (((((o))))), (((ooo))), ((oo(o))), (oo((o))), (o(o)(o)), (ooooo).

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000081, A000598, A003238, A004111, A027193, A032305, A067659, A290689, A291443, A297791, A298120.

orut[n_]:=orut[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[orut/@c]]]/@Select[IntegerPartitions[n-1],OddQ[Length[#]]&]];
Table[Length[orut[n]],{n,15}]

a(n) ~ c * d^n / n^(3/2), where d = 2.30984417428419893876754252289588812511559... and c = 0.5598122522173731208680575003383895445787... - Vaclav Kotesovec, Jun 04 2019

a(24)-a(35) from Alois P. Heinz, Jan 12 2018

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

A291442 Matula-Goebel numbers of leaf-balanced trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 27, 29, 30, 31, 32, 33, 36, 37, 40, 41, 44, 45, 47, 48, 49, 50, 53, 54, 55, 59, 60, 61, 62, 64, 66, 67, 71, 72, 75, 79, 80, 81, 83, 88, 89, 90, 91, 93, 96, 97, 99, 100, 103, 108

Offset: 1

Gus Wiseman, Aug 23 2017

nonn

An unlabeled rooted tree is leaf-balanced if every branch has the same number of leaves and every non-leaf rooted subtree is also leaf-balanced.

Cf. A000081, A007097, A061775, A109129, A214577, A280994, A291441, A291443.

nn=2000;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
leafcount[n_]:=If[n===1,1,With[{m=primeMS[n]},If[Length[m]===1,leafcount[First[m]],Total[leafcount/@m]]]];
balQ[n_]:=Or[n===1,With[{m=primeMS[n]},And[SameQ@@leafcount/@m,And@@balQ/@m]]];
Select[Range[nn],balQ]

A291441 Matula-Goebel numbers of orderless same-trees with all leaves equal to 1.

Original entry on oeis.org

1, 4, 8, 16, 32, 49, 64, 128, 256, 343, 361, 512, 1024, 2048, 2401, 2809, 4096, 6859, 8192, 12031, 16384, 16807, 17161, 32768, 51529, 65536, 96721, 117649, 130321, 131072, 148877, 262144, 516961, 524288, 637643, 718099, 757907, 823543, 1048576, 2097152, 2248091

Offset: 1

Gus Wiseman, Aug 23 2017

nonn

See A289078 for the definition of orderless same-tree.

			a(20)=12031 corresponds to the following same-tree: {{1,1,1,1},{{1,1},{1,1}}}.

Cf. A007097, A109129, A214577, A280994, A289078, A289079, A291442, A291443.

nn=200000;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
leafcount[n_]:=If[n===1,1,With[{m=primeMS[n]},If[Length[m]===1,leafcount[First[m]],Total[leafcount/@m]]]];
sameQ[n_]:=Or[n===1,With[{m=primeMS[n]},And[Length[m]>1,SameQ@@leafcount/@m,And@@sameQ/@m]]];
Select[Range[nn],sameQ]

More terms from Jinyuan Wang, Jun 21 2020

A295461 Number of unlabeled rooted trees with 2n + 1 nodes in which all outdegrees are even.

Original entry on oeis.org

1, 1, 2, 5, 12, 33, 91, 264, 780, 2365, 7274, 22727, 71784, 229094, 737215, 2390072, 7798020, 25587218, 84377881, 279499063, 929556155, 3102767833, 10390936382, 34903331506, 117564309276, 396994228503, 1343716120550, 4557952756658, 15491856887741

Offset: 0

Gus Wiseman, Jan 13 2018

nonn

			The a(3) = 5 trees: (o(o(oo))), (o(oooo)), ((oo)(oo)), (ooo(oo)), (oooooo).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000081, A000598, A001190, A003238, A004111, A027193, A027187, A032305, A067659, A290689, A291443, A297791, A298118, A298120, A298126.

erut[n_]:=erut[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[erut/@c]]]/@Select[IntegerPartitions[n-1],EvenQ[Length[#]]&]];
Table[Length[erut[n]],{n,1,30,2}]

A297571 Matula-Goebel numbers of fully unbalanced rooted trees.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 13, 15, 22, 26, 29, 30, 31, 33, 39, 41, 47, 55, 58, 62, 65, 66, 78, 79, 82, 87, 93, 94, 101, 109, 110, 113, 123, 127, 130, 137, 141, 145, 155, 158, 165, 167, 174, 179, 186, 195, 202, 205, 211, 218, 226, 235, 237, 246, 254, 257, 271, 274

Offset: 1

Gus Wiseman, Dec 31 2017

nonn

An unlabeled rooted tree is fully unbalanced if either (1) it is a single node, or (2a) every branch has a different number of nodes and (2b) every branch is fully unbalanced also. The number of fully unbalanced trees with n nodes is A032305(n).

The first finitary number (A276625) not in this sequence is 143.

			Sequence of fully unbalanced trees begins:
   1 o
   2 (o)
   3 ((o))
   5 (((o)))
   6 (o(o))
  10 (o((o)))
  11 ((((o))))
  13 ((o(o)))
  15 ((o)((o)))
  22 (o(((o))))
  26 (o(o(o)))
  29 ((o((o))))
  30 (o(o)((o)))
  31 (((((o)))))
  33 ((o)(((o))))
  39 ((o)(o(o)))
  41 (((o(o))))
  47 (((o)((o))))

Cf. A000081, A003238, A004111, A007097, A032305, A061775, A214577, A273873, A276625, A277098, A290689, A290760, A291441, A291442, A291443.

nn=2000;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
MGweight[n_]:=If[n===1,1,1+Total[Cases[FactorInteger[n],{p_,k_}:>k*MGweight[PrimePi[p]]]]];
imbalQ[n_]:=Or[n===1,With[{m=primeMS[n]},And[UnsameQ@@MGweight/@m,And@@imbalQ/@m]]];
Select[Range[nn],imbalQ]

A297791 Number of series-reduced leaf-balanced rooted trees with n nodes. Number of orderless same-trees with n nodes and all leaves equal to 1.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 5, 1, 3, 3, 4, 3, 5, 3, 6, 4, 6, 3, 12, 3, 10, 7, 9, 6, 12, 9, 13, 16, 14, 22, 22, 24, 21, 24, 28, 14, 32, 15, 42, 20, 60, 27, 84, 44, 100, 59, 113, 74, 116, 85, 110, 97, 96, 113, 106, 149, 147, 234, 235, 377, 380, 580, 576, 838

Offset: 1

Gus Wiseman, Jan 06 2018

nonn

An unlabeled rooted tree is leaf-balanced if all branches from the same root have the same number of leaves. It is series-reduced if all positive out-degrees are greater than one.

			The a(13) = 5 trees: (((oo)(oo))(oooo)), ((ooooo)(ooooo)), ((ooo)(ooo)(ooo)), ((oo)(oo)(oo)(oo)), (oooooooooooo).

Robert G. Wilson v, Table of n, a(n) for n = 1..84

Cf. A000081, A001190, A001678, A006241, A032305, A289078, A289079, A291441, A291443.

alltim[n_]:=alltim[n]=If[n===1,{{}},Join@@Function[c,Select[Union[Sort/@Tuples[alltim/@c]],And[SameQ@@(Count[#,{},{0,Infinity}]&/@#),FreeQ[#,{_}]]&]]/@IntegerPartitions[n-1]];
Table[Length[alltim[n]],{n,20}]

lista(nn) = my(k, r, t, u, w=vector(nn, i, vector(i))); w[1][1]=1; for(s=2, nn, fordiv(s, d, if(dw[i][d], [d..nn]); forvec(v=vector(s/d, i, [1, #u]), if(nn>=r=1+sum(i=1, #v, u[v[i]]), k=1; t=1; for(i=2, #v, if(v[i]==v[i-1], k++, t*=binomial(w[u[v[i-1]]][d]+k-1, k); k=1)); w[r][s]+=t*binomial(w[u[v[#v]]][d]+k-1, k)), 1)))); vector(nn, i, vecsum(w[i])); \\ Jinyuan Wang, Feb 25 2025

a(51) onward from Robert G. Wilson v, Jan 07 2018

A301343 Regular triangle where T(n,k) is the number of planted achiral (or generalized Bethe) trees with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 3, 2, 2, 1, 1, 0, 1, 3, 2, 2, 1, 1, 1, 0, 1, 4, 2, 4, 1, 2, 1, 1, 0, 1, 4, 3, 4, 1, 3, 1, 1, 1, 0, 1, 5, 3, 6, 2, 4, 1, 2, 1, 1, 0, 1, 5, 3, 6, 2, 4, 1, 2, 1, 1, 1, 0, 1, 6, 4, 9, 2, 7, 1, 4, 2, 2, 1, 1, 0

Offset: 1

Gus Wiseman, Mar 19 2018

			Triangle begins:
1
1  0
1  1  0
1  1  1  0
1  2  1  1  0
1  2  1  1  1  0
1  3  2  2  1  1  0
1  3  2  2  1  1  1  0
1  4  2  4  1  2  1  1  0
1  4  3  4  1  3  1  1  1  0
1  5  3  6  2  4  1  2  1  1  0
The T(9,4) = 4 planted achiral trees: (((((oooo))))), ((((oo)(oo)))), (((oo))((oo))), ((o)(o)(o)(o)).

Row sums are A003238. A version without the zeroes or first row is A214575.

Cf. A000081, A003238, A004111, A032305, A055277, A214577, A273873, A289078, A289079, A290689, A291443, A298422, A298426, A301342, A301344, A301345.

tri[n_,k_]:=If[k===1,1,If[k>=n,0,Sum[tri[n-k,d],{d,Divisors[k]}]]];
Table[tri[n,k],{n,10},{k,n}]

T(n,1) = 1, T(n,k) = 0 if n <= k, otherwise T(n,k) = Sum_{d|k} T(n - k, d).

A298533 Number of unlabeled rooted trees with n vertices such that every branch of the root has the same number of leaves.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 31, 64, 144, 333, 808, 2004, 5109, 13199, 34601, 91539, 244307, 656346, 1774212, 4820356, 13157591, 36060811, 99198470, 273790194, 757971757, 2104222594, 5856496542, 16338140048, 45678276507, 127964625782, 359155302204, 1009790944307

Offset: 1

Gus Wiseman, Jan 20 2018

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000081, A003238, A004111, A032305, A289079, A290689, A291443, A297791, A298422, A298534, A298535.

rut[n_]:=rut[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[rut/@c]]]/@IntegerPartitions[n-1]];
Table[Length[Select[rut[n],SameQ@@(Count[#,{},{0,Infinity}]&/@#)&]],{n,15}]

\\ here R is A055277 as vector of polynomials
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
R(n) = {my(A = O(x)); for(j=1, n, A = x*(y - 1  + exp( sum(i=1, j, 1/i * subst( subst( A + x * O(x^(j\i)), x, x^i), y, y^i) ) ))); Vec(A)};
seq(n)={my(M=Mat(apply(p->Colrev(p,n), R(n-1)))); concat([1],sum(i=2, #M, EulerT(M[i,])))} \\ Andrew Howroyd, May 20 2018

Terms a(19) and beyond from Andrew Howroyd, May 20 2018

cp's OEIS Frontend

A032305 Number of rooted trees where any 2 subtrees extending from the same node have a different number of nodes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A298118 Number of unlabeled rooted trees with n nodes in which all positive outdegrees are odd.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A291442 Matula-Goebel numbers of leaf-balanced trees.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A291441 Matula-Goebel numbers of orderless same-trees with all leaves equal to 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A295461 Number of unlabeled rooted trees with 2n + 1 nodes in which all outdegrees are even.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A297571 Matula-Goebel numbers of fully unbalanced rooted trees.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A297791 Number of series-reduced leaf-balanced rooted trees with n nodes. Number of orderless same-trees with n nodes and all leaves equal to 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs