A291636 -id:A291636 - cp's OEIS frontend

A331933 Number of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex.

1, 1, 1, 2, 4, 6, 12, 18, 33, 52, 90, 142, 242, 384, 639, 1028, 1688, 2716, 4445, 7161, 11665, 18839, 30595, 49434, 80199, 129637, 210079, 339750, 550228, 889978, 1440909, 2330887, 3772845, 6103823, 9878357, 15982196, 25863454, 41845650, 67713550, 109559443

Offset: 1

Gus Wiseman, Feb 03 2020

nonn

A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.

			The a(1) = 1 through a(8) = 18 trees:
  o  (o)  (oo)  (ooo)   (oooo)    (ooooo)    (oooooo)
                (o(o))  (o(oo))   (o(ooo))   (o(oooo))
                        (oo(o))   (oo(oo))   (oo(ooo))
                        ((o)(o))  (ooo(o))   (ooo(oo))
                                  (o(o)(o))  (oooo(o))
                                  (o(o(o)))  ((oo)(oo))
                                             (o(o(oo)))
                                             (o(oo(o)))
                                             (oo(o)(o))
                                             (oo(o(o)))
                                             ((o)(o)(o))
                                             (o((o)(o)))

Andrew Howroyd, Table of n, a(n) for n = 1..1000
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
Gus Wiseman, The a(11) = 90 semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex.

Not requiring lone-child-avoidance gives A320222.
The non-semi version is A320268.
Matula-Goebel numbers of these trees are A331936.
Achiral trees are A003238.
Semi-identity trees are A306200.
Numbers S with at most one distinct prime index in S are A331912.
Semi-lone-child-avoiding rooted trees are A331934.
Cf. A000081, A001678, A004111, A050381, A214577, A291636, A320230, A320269, A331784, A331872, A331914, A331935, A331966.

sseo[n_]:=Switch[n,1,{{}},2,{{{}}},_,Join@@Function[c,Select[Union[Sort/@Tuples[sseo/@c]],Length[Union[DeleteCases[#,{}]]]<=1&]]/@Rest[IntegerPartitions[n-1]]];
Table[Length[sseo[n]],{n,10}]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sum(i=2, n-2, ((n-1)\i)*v[i])); v} \\ Andrew Howroyd, Feb 09 2020

a(n) = 1 + Sum_{i=2..n-2} floor((n-1)/i)*a(i). - Andrew Howroyd, Feb 09 2020

Terms a(31) and beyond from Andrew Howroyd, Feb 09 2020

A298120 Matula-Goebel numbers of rooted trees in which all positive outdegrees are odd.

Original entry on oeis.org

1, 2, 3, 5, 8, 11, 12, 18, 19, 20, 27, 30, 31, 32, 37, 44, 45, 48, 50, 61, 66, 67, 71, 72, 75, 76, 80, 99, 103, 108, 110, 113, 114, 120, 124, 125, 127, 128, 131, 148, 157, 162, 165, 171, 176, 180, 186, 190, 192, 193, 197, 200, 222, 223, 229, 242, 243, 244, 264

Offset: 1

Gus Wiseman, Jan 12 2018

nonn

			Sequence of trees begins:
1  o
2  (o)
3  ((o))
5  (((o)))
8  (ooo)
11 ((((o))))
12 (oo(o))
18 (o(o)(o))
19 ((ooo))
20 (oo((o)))
27 ((o)(o)(o))
30 (o(o)((o)))
31 (((((o)))))
32 (ooooo)
37 ((oo(o)))
44 (oo(((o))))
45 ((o)(o)((o)))
48 (oooo(o))
50 (o((o))((o)))

Index entries for sequences related to Matula-Goebel numbers

Cf. A000081, A007097, A026424, A061775, A214577, A276625, A277098, A290760, A291441, A291442, A291636, A297571, A298118.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
otQ[n_]:=Or[n===1,With[{m=primeMS[n]},OddQ@Length@m&&And@@otQ/@m]];
Select[Range[1000],otQ]

A331488 Number of unlabeled lone-child-avoiding rooted trees with n vertices and more than two branches (of the root).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 6, 10, 20, 36, 70, 134, 263, 513, 1022, 2030, 4076, 8203, 16614, 33738, 68833, 140796, 288989, 594621, 1226781, 2536532, 5256303, 10913196, 22700682, 47299699, 98714362, 206323140, 431847121, 905074333, 1899247187, 3990145833, 8392281473

Offset: 1

Gus Wiseman, Jan 20 2020

nonn

Also the number of lone-child-avoiding rooted trees with n vertices and more than two branches.

			The a(4) = 1 through a(9) = 10 trees:
  (ooo)  (oooo)  (ooooo)   (oooooo)   (ooooooo)    (oooooooo)
                 (oo(oo))  (oo(ooo))  (oo(oooo))   (oo(ooooo))
                           (ooo(oo))  (ooo(ooo))   (ooo(oooo))
                                      (oooo(oo))   (oooo(ooo))
                                      (o(oo)(oo))  (ooooo(oo))
                                      (oo(o(oo)))  (o(oo)(ooo))
                                                   (oo(o(ooo)))
                                                   (oo(oo)(oo))
                                                   (oo(oo(oo)))
                                                   (ooo(o(oo)))

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014)
Eric Weisstein's World of Mathematics, Series-reduced tree.
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

The not necessarily lone-child-avoiding version is A331233.
The Matula-Goebel numbers of these trees are listed by A331490.
A000081 counts unlabeled rooted trees.
A001678 counts lone-child-avoiding rooted trees.
A001679 counts topologically series-reduced rooted trees.
A291636 lists Matula-Goebel numbers of lone-child-avoiding rooted trees.
A331489 lists Matula-Goebel numbers of series-reduced rooted trees.
Cf. A000014, A000669, A004250, A007097, A007821, A033942, A060313, A060356, A061775, A109082, A109129, A196050, A276625, A330943.

urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[Length[Select[urt[n],Length[#]>2&&FreeQ[#,{_}]&]],{n,10}]

For n > 1, a(n) = A001679(n) - A001678(n).

a(37)-a(38) from Jinyuan Wang, Jun 26 2020

Terminology corrected (lone-child-avoiding, not series-reduced) by Gus Wiseman, May 10 2021

A331871 Matula-Goebel numbers of lone-child-avoiding locally disjoint rooted trees.

Original entry on oeis.org

1, 4, 8, 14, 16, 28, 32, 38, 49, 56, 64, 76, 86, 98, 106, 112, 128, 152, 172, 196, 212, 214, 224, 256, 262, 304, 326, 343, 344, 361, 392, 424, 428, 448, 454, 512, 524, 526, 608, 622, 652, 686, 688, 722, 766, 784, 848, 856, 886, 896, 908, 1024, 1042, 1048, 1052

Offset: 1

Gus Wiseman, Feb 02 2020

nonn

First differs from A320269 in having 1589, the Matula-Goebel number of the tree ((oo)((oo)(oo))).
First differs from A331683 in having 49.
A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex.
Lone-child-avoiding means there are no unary branchings.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
Consists of one and all nonprime numbers whose distinct prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be pairwise coprime. A prime index of n is a number m such that prime(m) divides n.

			The sequence of all lone-child-avoiding locally disjoint rooted trees together with their Matula-Goebel numbers begins:
    1: o
    4: (oo)
    8: (ooo)
   14: (o(oo))
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   38: (o(ooo))
   49: ((oo)(oo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
   98: (o(oo)(oo))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  152: (ooo(ooo))
  172: (oo(o(oo)))
  196: (oo(oo)(oo))
The sequence of terms together with their prime indices begins:
     1: {}                  212: {1,1,16}
     4: {1,1}               214: {1,28}
     8: {1,1,1}             224: {1,1,1,1,1,4}
    14: {1,4}               256: {1,1,1,1,1,1,1,1}
    16: {1,1,1,1}           262: {1,32}
    28: {1,1,4}             304: {1,1,1,1,8}
    32: {1,1,1,1,1}         326: {1,38}
    38: {1,8}               343: {4,4,4}
    49: {4,4}               344: {1,1,1,14}
    56: {1,1,1,4}           361: {8,8}
    64: {1,1,1,1,1,1}       392: {1,1,1,4,4}
    76: {1,1,8}             424: {1,1,1,16}
    86: {1,14}              428: {1,1,28}
    98: {1,4,4}             448: {1,1,1,1,1,1,4}
   106: {1,16}              454: {1,49}
   112: {1,1,1,1,4}         512: {1,1,1,1,1,1,1,1,1}
   128: {1,1,1,1,1,1,1}     524: {1,1,32}
   152: {1,1,1,8}           526: {1,56}
   172: {1,1,14}            608: {1,1,1,1,1,8}
   196: {1,1,4,4}           622: {1,64}

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Not requiring local disjointness gives A291636.
Not requiring lone-child avoidance gives A316495.
A superset of A320269.
These trees are counted by A331680.
The semi-identity tree version is A331683.
The version containing 2 is A331873.
Cf. A001678, A007097, A050381, A061775, A196050, A302569, A302696, A316473, A316694, A316696, A316697, A331682, A331686, A331687, A331872, A331935.

msQ[n_]:=n==1||!PrimeQ[n]&&(PrimePowerQ[n]||CoprimeQ@@PrimePi/@First/@FactorInteger[n])&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[1000],msQ]

Intersection of A291636 and A316495.

A301344 Regular triangle where T(n,k) is the number of semi-binary rooted trees with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 4, 1, 0, 0, 1, 6, 4, 0, 0, 0, 1, 9, 11, 2, 0, 0, 0, 1, 12, 24, 9, 0, 0, 0, 0, 1, 16, 46, 32, 3, 0, 0, 0, 0, 1, 20, 80, 86, 20, 0, 0, 0, 0, 0, 1, 25, 130, 203, 86, 6, 0, 0, 0, 0, 0, 1, 30, 200, 423, 283, 46, 0, 0, 0, 0, 0, 0, 1, 36, 295, 816, 786, 234, 11, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Mar 19 2018

A rooted tree is semi-binary if all outdegrees are <= 2. The number of semi-binary trees with n nodes is equal to the number of binary trees with n+1 leaves; see A001190.

			Triangle begins:
1
1   0
1   1   0
1   2   0   0
1   4   1   0   0
1   6   4   0   0   0
1   9  11   2   0   0   0
1  12  24   9   0   0   0   0
1  16  46  32   3   0   0   0   0
1  20  80  86  20   0   0   0   0   0
1  25 130 203  86   6   0   0   0   0   0
The T(6,3) = 4 semi-binary rooted trees: ((o(oo))), (o((oo))), (o(o(o))), ((o)(oo)).

Cf. A000081, A000598, A001190, A001678, A003238, A004111, A055277, A111299, A273873, A290689, A291636, A292050, A298118, A298204, A298422, A298426, A301342, A301343, A301345.

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

A318186 Totally transitive numbers. Matula-Goebel numbers of totally transitive rooted trees.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 16, 18, 24, 28, 32, 36, 38, 42, 48, 54, 56, 64, 72, 76, 78, 84, 96, 98, 106, 108, 112, 114, 126, 128, 144, 152, 156, 162, 168, 192, 196, 212, 216, 222, 224, 228, 234, 252, 256, 262, 266, 288, 294, 304, 312, 318, 324, 336, 342, 366, 378

Offset: 1

Gus Wiseman, Aug 20 2018

nonn

A number x is totally transitive if (1) whenever prime(y) divides x it follows that y is totally transitive and (2) if prime(y) divides x and prime(z) divides y then prime(z) also divides x.

			The sequence of all totally transitive rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  18: (o(o)(o))
  24: (ooo(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  42: (o(o)(oo))
  48: (oooo(o))
  54: (o(o)(o)(o))
  56: (ooo(oo))
  64: (oooooo)
  72: (ooo(o)(o))
  76: (oo(ooo))
  78: (o(o)(o(o)))
  84: (oo(o)(oo))
  96: (ooooo(o))
  98: (o(oo)(oo))

Cf. A000081, A001678, A004111, A007097, A061775, A276625, A279861, A290689, A290760, A290822, A291636, A318185, A318187.

subprimes[n_]:=If[n==1,{},Union@@Cases[FactorInteger[n],{p_,_}:>FactorInteger[PrimePi[p]][[All,1]]]];
trmgQ[n_]:=Or[n==1,And[Divisible[n,Times@@subprimes[n]],And@@Cases[FactorInteger[n],{p_,_}:>trmgQ[PrimePi[p]]]]];
Select[Range[100],trmgQ]

A319376 Triangle read by rows: T(n,k) is the number of lone-child-avoiding rooted trees with n leaves of exactly k colors.

Original entry on oeis.org

1, 1, 1, 2, 6, 4, 5, 30, 51, 26, 12, 146, 474, 576, 236, 33, 719, 3950, 8572, 8060, 2752, 90, 3590, 31464, 108416, 175380, 134136, 39208, 261, 18283, 245916, 1262732, 3124650, 4014348, 2584568, 660032, 766, 94648, 1908858, 14047288, 49885320, 95715728, 101799712, 56555904, 12818912

Offset: 1

Andrew Howroyd, Sep 17 2018

Lone-child-avoiding rooted trees are also called planted series-reduced trees in some other sequences.

			Triangle begins:
    1;
    1,     1;
    2,     6,      4;
    5,    30,     51,      26;
   12,   146,    474,     576,     236;
   33,   719,   3950,    8572,    8060,   2752;
   90,  3590,  31464,  108416,  175380,  134136,   39208;
  261, 18283, 245916, 1262732, 3124650, 4014348, 2584568, 660032;
  ...
From _Gus Wiseman_, Dec 31 2020: (Start)
The 12 trees counted by row n = 3:
  (111)    (112)    (123)
  (1(11))  (122)    (1(23))
           (1(12))  (2(13))
           (1(22))  (3(12))
           (2(11))
           (2(12))
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Columns k=1..2 are A000669, A319377.
Main diagonal is A000311.
Row sums are A316651.
Cf. A141610, A242249, A255517, A256064, A256068, A319254, A319541, A326396.
The unlabeled version, counting inequivalent leaf-colorings of lone-child-avoiding rooted trees, is A330465.
Lone-child-avoiding rooted trees are counted by A001678 (shifted left once).
Labeled lone-child-avoiding rooted trees are counted by A060356.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Cf. A000014, A000081, A000169, A005804, A206429, A330951.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(A(i, k)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
A:= (n, k)-> `if`(n<2, n*k, b(n, n-1, k)):
T:= (n, k)-> add(A(n, k-j)*(-1)^j*binomial(k, j), j=0..k-1):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Sep 18 2018

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[A[i, k] + j - 1, j] b[n - i j, i - 1, k], {j, 0, n/i}]]];
A[n_, k_] := If[n < 2, n k, b[n, n - 1, k]];
T[n_, k_] := Sum[(-1)^(k - i)*Binomial[k, i]*A[n, i], {i, 1, k}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 24 2019, after Alois P. Heinz *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
mtot[m_]:=Prepend[Join@@Table[Tuples[mtot/@p],{p,Select[mps[m],1Gus Wiseman, Dec 31 2020 *)

\\ here R(n,k) is k-th column of A319254.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[k]); for(n=2, n, v=concat(v, EulerT(concat(v, [0]))[n])); v}
M(n)={my(v=vector(n, k, R(n,k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k,i)*v[i])))}
{my(T=M(10)); for(n=1, #T~, print(T[n, ][1..n]))}

T(n,k) = Sum_{i=1..k} (-1)^(k-i)*binomial(k,i)*A319254(n,i).

Sum_{k=1..n} k * T(n,k) = A326396(n). - Alois P. Heinz, Sep 11 2019

A331490 Matula-Goebel numbers of series-reduced rooted trees with more than two branches (of the root).

Original entry on oeis.org

8, 16, 28, 32, 56, 64, 76, 98, 112, 128, 152, 172, 196, 212, 224, 256, 266, 304, 343, 344, 392, 424, 428, 448, 512, 524, 532, 602, 608, 652, 686, 688, 722, 742, 784, 848, 856, 896, 908, 931, 1024, 1048, 1052, 1064, 1204, 1216, 1244, 1304, 1372, 1376, 1444

Offset: 1

Gus Wiseman, Jan 20 2020

nonn

We say that a rooted tree is (topologically) series-reduced if no vertex has degree 2.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches. This gives a bijective correspondence between positive integers and unlabeled rooted trees.
Also Matula-Goebel numbers of lone-child-avoiding rooted trees with more than two branches.

			The sequence of all series-reduced rooted trees with more than two branches together with their Matula-Goebel numbers begins:
    8: (ooo)
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   98: (o(oo)(oo))
  112: (oooo(oo))
  128: (ooooooo)
  152: (ooo(ooo))
  172: (oo(o(oo)))
  196: (oo(oo)(oo))
  212: (oo(oooo))
  224: (ooooo(oo))
  256: (oooooooo)
  266: (o(oo)(ooo))
  304: (oooo(ooo))
  343: ((oo)(oo)(oo))
  344: (ooo(o(oo)))

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014)
Eric Weisstein's World of Mathematics, Series-reduced tree.
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

These trees are counted by A331488.
Unlabeled rooted trees are counted by A000081.
Lone-child-avoiding rooted trees are counted by A001678.
Topologically series-reduced rooted trees are counted by A001679.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Matula-Goebel numbers of series-reduced rooted trees are A331489.
Cf. A000014, A000669, A004250, A007097, A007821, A033942, A060313, A060356, A061775, A109082, A109129, A196050, A276625, A330943.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
srQ[n_]:=Or[n==1,With[{m=primeMS[n]},And[Length[m]>1,And@@srQ/@m]]];
Select[Range[1000],PrimeOmega[#]>2&&srQ[#]&]

A358376 Numbers k such that the k-th standard ordered rooted tree is lone-child-avoiding (counted by A005043).

Original entry on oeis.org

1, 4, 8, 16, 18, 25, 32, 36, 50, 57, 64, 72, 100, 114, 121, 128, 137, 144, 200, 228, 242, 249, 256, 258, 274, 281, 288, 385, 393, 400, 456, 484, 498, 505, 512, 516, 548, 562, 569, 576, 770, 786, 793, 800, 897, 905, 912, 968, 996, 1010, 1017, 1024, 1032, 1096

Offset: 1

Gus Wiseman, Nov 14 2022

nonn

We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

			The initial terms and their corresponding trees:
    1: o
    4: (oo)
    8: (ooo)
   16: (oooo)
   18: ((oo)o)
   25: (o(oo))
   32: (ooooo)
   36: ((oo)oo)
   50: (o(oo)o)
   57: (oo(oo))
   64: (oooooo)
   72: ((oo)ooo)
  100: (o(oo)oo)
  114: (oo(oo)o)
  121: (ooo(oo))
  128: (ooooooo)
  137: ((oo)(oo))
  144: ((oo)oooo)
  200: (o(oo)ooo)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

These trees are counted by A005043.
The series-reduced case appears to be counted by A284778.
The unordered version is A291636, counted by A001678.
A000081 counts unlabeled rooted trees, ranked by A358378.
A358371 and A358372 count leaves and nodes in standard ordered rooted trees.
A358374 ranks ordered identity trees, counted by A032027.
A358375 ranks ordered binary trees, counted by A126120.
Cf. A000014, A001263, A001679, A004249, A061775, A063895, A187306, A331489, A331490, A331934, A358373, A358377.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Select[Range[100],FreeQ[srt[#],[_]?(Length[#]==1&)]&]

A254382 Number of rooted labeled trees on n nodes such that every nonroot node is the child of a branching node or of the root.

Original entry on oeis.org

0, 1, 2, 3, 16, 85, 696, 6349, 72080, 918873, 13484080, 219335281, 3962458248, 78203547877, 1680235050872, 38958029188485, 970681471597216, 25847378934429361, 732794687650764000, 22032916968153975769, 700360446794528578520

Offset: 0

Geoffrey Critzer, Jan 29 2015

nonn

Here, a branching node is a node with at least two children.
In other words, a(n) is the number of labeled rooted trees on n nodes such that the path from every node towards the root reaches a branching node (or the root) in one step.
Also labeled rooted trees that are lone-child-avoiding except possibly for the root. The unlabeled version is A198518. - Gus Wiseman, Jan 22 2020

			a(5) = 85:
...0................0...............0-o...
...|.............../ \............ /|\....
...o..............o   o...........o o o...
../|\............/ \   ...................
.o o o..........o   o   ..................
These trees have 20 + 60 + 5 = 85 labelings.
From _Gus Wiseman_, Jan 22 2020: (Start)
The a(1) = 1 through a(4) = 16 trees (in the format root[branches]) are:
  1  1[2]  1[2,3]  1[2,3,4]
     2[1]  2[1,3]  1[2[3,4]]
           3[1,2]  1[3[2,4]]
                   1[4[2,3]]
                   2[1,3,4]
                   2[1[3,4]]
                   2[3[1,4]]
                   2[4[1,3]]
                   3[1,2,4]
                   3[1[2,4]]
                   3[2[1,4]]
                   3[4[1,2]]
                   4[1,2,3]
                   4[1[2,3]]
                   4[2[1,3]]
                   4[3[1,2]]
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..200
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Cf. A231797, A052318 (condition is applied only to leaf nodes).
The unlabeled version is A198518
The non-planted case is A060356.
Labeled rooted trees are A000169.
Lone-child-avoiding rooted trees are A001678(n + 1).
Labeled topologically series-reduced rooted trees are A060313.
Labeled lone-child-avoiding unrooted trees are A108919.
Cf. A000014, A000669, A001679, A005512, A291636, A331488, A331578.

nn = 20; b = 1 + Sum[nn = n; n! Coefficient[Series[(Exp[x] - x)^n, {x, 0, nn}], x^n]*x^n/n!, {n,1, nn}]; c = Sum[a[n] x^n/n!, {n, 0, nn}]; sol = SolveAlways[b == Series[1/(1 - (c - x)), {x, 0, nn}], x]; Flatten[Table[a[n], {n, 0, nn}] /. sol]
nn = 30; CoefficientList[Series[1+x-1/Sum[SeriesCoefficient[(E^x-x)^n,{x,0,n}]*x^n,{n,0,nn}],{x,0,nn}],x] * Range[0,nn]! (* Vaclav Kotesovec, Jan 30 2015 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
lrt[set_]:=If[Length[set]==0,{},Join@@Table[Apply[root,#]&/@Join@@Table[Tuples[lrt/@stn],{stn,sps[DeleteCases[set,root]]}],{root,set}]];
Table[Length[Select[lrt[Range[n]],FreeQ[Z@@#,Integer[]]&]],{n,6}] (* Gus Wiseman, Jan 22 2020 *)

E.g.f.: A(x) satisfies 1/(1 - (A(x) - x)) = B(x) where B(x) is the e.g.f. for A231797.

a(n) ~ (1-exp(-1))^(n-1/2) * n^(n-1). - Vaclav Kotesovec, Jan 30 2015

cp's OEIS Frontend

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