A032305 -id:A032305 - cp's OEIS frontend

A318754 Number T(n,k) of rooted trees with n nodes such that k equals the maximal number of subtrees extending from the same node and having the same number of nodes; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 3, 4, 1, 1, 0, 6, 9, 3, 1, 1, 0, 12, 22, 9, 3, 1, 1, 0, 25, 54, 23, 8, 3, 1, 1, 0, 51, 139, 60, 23, 8, 3, 1, 1, 0, 111, 346, 166, 61, 22, 8, 3, 1, 1, 0, 240, 892, 447, 167, 61, 22, 8, 3, 1, 1, 0, 533, 2290, 1219, 461, 168, 60, 22, 8, 3, 1, 1

Offset: 1

Alois P. Heinz, Sep 02 2018

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k < n. T(n,k) = 0 for k >= n.

			Triangle T(n,k) begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   1,   1;
  0,   3,   4,   1,  1;
  0,   6,   9,   3,  1,  1;
  0,  12,  22,   9,  3,  1, 1;
  0,  25,  54,  23,  8,  3, 1, 1;
  0,  51, 139,  60, 23,  8, 3, 1, 1;
  0, 111, 346, 166, 61, 22, 8, 3, 1, 1;

Alois P. Heinz, Rows n = 1..200, flattened

Columns k=0-10 give: A063524, A032305 (for n>1), A318817, A318818, A318819, A318820, A318821, A318822, A318823, A318824, A318825.
Row sums give A000081.
T(2n+2,n+1) give A255705.
Cf. A318753.

g:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(g(i-1$2, k)+j-1, j)*g(n-i*j, i-1, k), j=0..min(k, n/i))))
    end:
T:= (n, k)-> g(n-1$2, k) -`if`(k=0, 0, g(n-1$2, k-1)):
seq(seq(T(n, k), k=0..n-1), n=1..14);

g[n_, i_, k_] := g[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[g[i - 1, i - 1, k] + j - 1, j]*g[n - i*j, i - 1, k], {j, 0, Min[k, n/i]}]]];
T[n_, k_] := g[n - 1, n - 1, k] - If[k == 0, 0, g[n - 1, n - 1, k - 1]];
Table[T[n, k], {n, 1, 14}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, May 27 2019, after Alois P. Heinz *)

T(n,k) = A318753(n,k) - A318753(n,k-1) for k > 0, A(n,0) = A063524(n).

A298204 Number of unlabeled rooted trees with n nodes in which all outdegrees are either 0, 1, or 3.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 16, 29, 55, 104, 200, 389, 763, 1507, 3002, 6010, 12102, 24484, 49751, 101475, 207723, 426542, 878451, 1813945, 3754918, 7790326, 16196629, 33739335, 70410401, 147187513, 308171861, 646188276, 1356847388, 2852809425, 6005542176

Offset: 1

Gus Wiseman, Jan 14 2018

nonn

			The a(7) = 9 trees: ((((((o)))))), ((((ooo)))), (((oo(o)))), ((oo((o)))), ((o(o)(o))), (oo(((o)))), (oo(ooo)), (o(o)((o))), ((o)(o)(o)).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Gus Wiseman, Illustration of the first 8 terms.

Cf. A000081, A000598, A001190, A001399, A004111, A014591, A032305, A273873, A292050, A295461, A298118, A298205.

b:= proc(n, i, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n `if`(n<2, n, add(b(n-1$2, j), j=[1, 3])):
seq(a(n), n=1..40);  # Alois P. Heinz, Jan 30 2018

multing[n_,k_]:=Binomial[n+k-1,k];
a[n_]:=a[n]=If[n===1,1,Sum[Product[multing[a[x],Count[ptn,x]],{x,Union[ptn]}],{ptn,Select[IntegerPartitions[n-1],MemberQ[{1,3},Length[#]]&]}]];
Table[a[n],{n,40}]
(* Second program: *)
b[n_, i_, v_] := b[n, i, v] = If[n == 0,
     If[v == 0, 1, 0], If[i < 1 || v < 1 || n < v, 0,
     If[n == v, 1, Sum[Binomial[a[i] + j - 1, j]*
     b[n - i*j, i - 1, v - j], {j, 0, Min[n/i, v]}]]]];
a[n_] := If[n < 2, n, Sum[b[n - 1, n - 1, j], {j, {1, 3}}]];
Array[a, 40] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

A301345 Regular triangle where T(n,k) is the number of transitive rooted trees with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 1, 2, 4, 1, 0, 0, 0, 0, 3, 4, 5, 1, 0, 0, 0, 0, 2, 6, 6, 6, 1, 0, 0, 0, 0, 1, 6, 10, 9, 7, 1, 0, 0, 0, 0, 1, 5, 12, 16, 12, 8, 1, 0, 0, 0, 0, 0, 4, 13, 22, 23, 16, 9, 1, 0, 0, 0, 0, 0, 3, 14, 27, 36, 32, 20, 10, 1, 0, 0, 0, 0, 0, 2, 11

Offset: 1

Gus Wiseman, Mar 19 2018

			Triangle begins:
1
1   0
0   1   0
0   1   1   0
0   0   2   1   0
0   0   1   3   1   0
0   0   1   2   4   1   0
0   0   0   3   4   5   1   0
0   0   0   2   6   6   6   1   0
0   0   0   1   6  10   9   7   1   0
0   0   0   1   5  12  16  12   8   1   0
The T(9,5) = 6 transitive rooted trees: (o(o)(oo(o))), (o((oo))(oo)), (oo(o)(o(o))), (o(o)(o)(oo)), (ooo(o)((o))), (oo(o)(o)(o)).

Row sums are A290689.

Cf. A000081, A001190, A003238, A004111, A032305, A055277, A276625, A279861, A290760, A290822, A298422, A298426, A301342, A301343, A301344.

rut[n_]:=rut[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[rut/@c]]]/@IntegerPartitions[n-1]];
trat[n_]:=Select[rut[n],Complement[Union@@#,#]==={}&];
Table[Length[Select[trat[n],Count[#,{},{-2}]===k&]],{n,15},{k,n}]

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).

A298304 Number of rooted trees on n nodes with strictly thinning limbs.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 12, 19, 31, 51, 85, 144, 245, 417, 712, 1221, 2091, 3600, 6216, 10763, 18691, 32546, 56782, 99271, 173849, 304877, 535412, 941385, 1657069, 2919930, 5150546, 9093894, 16071634, 28428838, 50331137, 89181251, 158145233, 280650225, 498410197

Offset: 1

Gus Wiseman, Jan 16 2018

nonn

An unlabeled rooted tree has strictly thinning limbs if its outdegrees are strictly decreasing from root to leaves.

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

Alois P. Heinz, Table of n, a(n) for n = 1..300

Cf. A000081, A001678, A004111, A032305, A124343, A290689, A295461, A298118, A298303, A298305.

stinctQ[t_]:=And@@Cases[t,b_List:>Length[b]>Max@@Length/@b,{0,Infinity}];
strut[n_]:=strut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@IntegerPartitions[n-1],stinctQ]];
Table[Length[strut[n]],{n,20}]

a(26)-a(40) from Alois P. Heinz, Jan 17 2018

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

A213920 Number of rooted trees with n nodes such that no more than two subtrees corresponding to children of any node have the same number of nodes.

Original entry on oeis.org

0, 1, 1, 2, 3, 7, 15, 34, 79, 190, 457, 1132, 2823, 7126, 18136, 46541, 120103, 312109, 815012, 2137755, 5632399, 14895684, 39519502, 105198371, 280815067, 751490363, 2016142768, 5420945437, 14604580683, 39425557103, 106618273626, 288792927325, 783516425820

Offset: 0

Alois P. Heinz, Mar 05 2013

nonn

Coincides with A248869 up to a(9) = 190.

a(n+1)/a(n) tends to 2.845331... - Vaclav Kotesovec, Jun 04 2019

			:  o  :  o  :    o   o  :    o     o   o  :
:     :  |  :   / \  |  :    |    / \  |  :
:     :  o  :  o   o o  :    o   o   o o  :
:     :     :        |  :   / \  |     |  :
:     :     :        o  :  o   o o     o  :
:     :     :           :              |  :
: n=1 : n=2 :  n=3      :  n=4         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 :
:                                       | :
: n=5                                   o :
:.........................................:

Alois P. Heinz, Table of n, a(n) for n = 0..2213

Cf. A000081, A032305, A248869.

Column k=2 of A318753.

g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(g((i-1)$2)+j-1, j)*g(n-i*j, i-1), j=0..min(2, n/i))))
    end:
a:= n-> g((n-1)$2):
seq(a(n), n=0..40);

g[n_, i_] := g[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[g[i-1, i-1]+j-1, j]*g[n-i*j, i-1], {j, 0, Min[2, n/i]}]]]; a[n_] := g[n-1, n-1]; Table[ a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 21 2017, translated from Maple *)

cp's OEIS Frontend

A318754 Number T(n,k) of rooted trees with n nodes such that k equals the maximal number of subtrees extending from the same node and having the same number of nodes; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

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A298204 Number of unlabeled rooted trees with n nodes in which all outdegrees are either 0, 1, or 3.

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A301345 Regular triangle where T(n,k) is the number of transitive rooted trees with n nodes and k leaves.

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Mathematica

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

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Mathematica

A297571 Matula-Goebel numbers of fully unbalanced rooted trees.

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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.

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

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Mathematica

Formula

A298304 Number of rooted trees on n nodes with strictly thinning limbs.

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