A004111 -id:A004111 - 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

A298422 Number of rooted trees with n nodes in which all positive outdegrees are the same.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 6, 4, 9, 2, 20, 2, 26, 12, 53, 2, 120, 2, 223, 43, 454, 2, 1100, 11, 2182, 215, 4902, 2, 11446, 2, 24744, 1242, 56014, 58, 131258, 2, 293550, 7643, 676928, 2, 1582686, 2, 3627780, 49155, 8436382, 2, 19809464, 50, 46027323, 321202

Offset: 1

Gus Wiseman, Jan 19 2018

nonn

Row sums of A298426.

			The a(9) = 6 trees: ((((((((o)))))))), (o(o(o(oo)))), (o((oo)(oo))), ((oo)(o(oo))), (ooo(oooo)), (oooooooo).

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A000005, A000081, A000598, A001190, A001678, A003238, A004111, A008864, A032305, A067538, A111299, A124343, A143773, A289078, A289079, A295461, A298118, A298204, A298423, A298424, A298426.

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

a(n) = 2 <=> n in {A008864}. - Alois P. Heinz, Jan 20 2018

a(44)-a(52) from Alois P. Heinz, Jan 20 2018

A301700 Number of aperiodic rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 21, 52, 120, 290, 697, 1713, 4200, 10446, 26053, 65473, 165257, 419357, 1068239, 2732509, 7013242, 18059960, 46641983, 120790324, 313593621, 816046050, 2128101601, 5560829666, 14557746453, 38177226541, 100281484375, 263815322761, 695027102020

Offset: 1

Gus Wiseman, Apr 23 2018

nonn

An unlabeled rooted tree is aperiodic if the multiset of branches of the root is an aperiodic multiset, meaning it has relatively prime multiplicities, and each branch is also aperiodic.

			The a(6) = 10 aperiodic trees are (((((o))))), (((o(o)))), ((o((o)))), ((oo(o))), (o(((o)))), (o(o(o))), ((o)((o))), (oo((o))), (o(o)(o)), (ooo(o)).

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

Cf. A000081, A000740, A000837, A001678, A003238, A004111, A007716, A007916, A100953, A276625, A284639, A290689, A298422, A303386, A303431.

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

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
MoebiusT(v)={vector(#v, n, sumdiv(n,d,moebius(n/d)*v[d]))}
seq(n)={my(v=[1]); for(n=2, n, v=concat([1], MoebiusT(EulerT(v)))); v} \\ Andrew Howroyd, Sep 01 2018

Terms a(21) and beyond from Andrew Howroyd, Sep 01 2018

A111299 Numbers whose Matula tree is a binary tree (i.e., root has degree 2 and all nodes except root and leaves have degree 3).

Original entry on oeis.org

4, 14, 49, 86, 301, 454, 886, 1589, 1849, 3101, 3986, 6418, 9761, 13766, 13951, 19049, 22463, 26798, 31754, 48181, 51529, 57026, 75266, 85699, 93793, 100561, 111139, 128074, 137987, 196249, 199591, 203878, 263431, 295969, 298154, 302426, 426058, 448259, 452411

Offset: 1

Keith Briggs, Nov 02 2005

nonn

This sequence should probably start with 1. Then a number k is in the sequence iff k = 1 or k = prime(x) * prime(y) with x and y already in the sequence. - Gus Wiseman, May 04 2021

			From _Gus Wiseman_, May 04 2021: (Start)
The sequence of trees (starting with 1) begins:
     1: o
     4: (oo)
    14: (o(oo))
    49: ((oo)(oo))
    86: (o(o(oo)))
   301: ((oo)(o(oo)))
   454: (o((oo)(oo)))
   886: (o(o(o(oo))))
  1589: ((oo)((oo)(oo)))
  1849: ((o(oo))(o(oo)))
  3101: ((oo)(o(o(oo))))
  3986: (o((oo)(o(oo))))
  6418: (o(o((oo)(oo))))
  9761: ((o(oo))((oo)(oo)))
(End)

Kevin Ryde, Table of n, a(n) for n = 1..5000 (terms 1..186 from Charles R Greathouse IV)
Keith Briggs, Matula numbers and rooted trees
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Kevin Ryde, PARI/GP Code
Index entries for sequences related to Matula-Goebel numbers

Cf. A245824 (by number of leaves).
Cf. A005517, A005518, A061773, A245824.
These trees are counted by 2*A001190 - 1.
The semi-binary version is A292050 (counted by A001190).
The semi-identity case is A339193 (counted by A063895).
A000081 counts unlabeled rooted trees with n nodes.
A007097 ranks rooted chains.
A276625 ranks identity trees, counted by A004111.
A306202 ranks semi-identity trees, counted by A306200.
A306203 ranks balanced semi-identity trees, counted by A306201.
A331965 ranks lone-child avoiding semi-identity trees, counted by A331966.
Cf. A036656, A061775, A196050, A291636, A320230, A331935, A331965.

nn=20000;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
binQ[n_]:=Or[n===1,With[{m=primeMS[n]},And[Length[m]===2,And@@binQ/@m]]];
Select[Range[2,nn],binQ] (* Gus Wiseman, Aug 28 2017 *)

i(n)=n==2 || is(primepi(n))
is(n)=if(n<14,return(n==4)); my(f=factor(n),t=#f[,1]); if(t>1, t==2 && f[1,2]==1 && f[2,2]==1 && i(f[1,1]) && i(f[2,1]), f[1,2]==2 && i(f[1,1])) \\ Charles R Greathouse IV, Mar 29 2013

list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, if(i(p)&&i(q), listput(v, t*q)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Mar 29 2013

PARI
```
\\ Also see links.
```

The Matula tree of k is defined as follows:
matula(k):
    create a node labeled k
    for each prime factor m of k:
       add the subtree matula(prime(m)), by an edge labeled m
    return the node

Definition corrected by Charles R Greathouse IV, Mar 29 2013

a(27)-a(39) from Charles R Greathouse IV, Mar 29 2013

A299757 Weight of the strict integer partition with FDH number n.

Original entry on oeis.org

0, 1, 2, 3, 4, 3, 5, 4, 6, 5, 7, 5, 8, 6, 6, 9, 10, 7, 11, 7, 7, 8, 12, 6, 13, 9, 8, 8, 14, 7, 15, 10, 9, 11, 9, 9, 16, 12, 10, 8, 17, 8, 18, 10, 10, 13, 19, 11, 20, 14, 12, 11, 21, 9, 11, 9, 13, 15, 22, 9, 23, 16, 11, 12, 12, 10, 24, 13, 14, 10, 25, 10, 26, 17

Offset: 1

Gus Wiseman, Feb 18 2018

nonn

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. This determines a unique strict integer partition (s_k...s_1) whose FDH number is then defined to be n.

In analogy with the Heinz number correspondence between integer partitions and positive integers (see A056239), FDH numbers give a correspondence between strict integer partitions and positive integers.

			Sequence of strict integer partitions begins: () (1) (2) (3) (4) (2,1) (5) (3,1) (6) (4,1) (7) (3,2) (8) (5,1) (4,2) (9).

Cf. A004111, A050376, A056239, A061775, A064547, A106400, A213925, A215366, A246867, A279065, A279614, A299090, A299755, A299756, A299758, A299759.

FDfactor[n_]:=If[n===1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
nn=200;FDprimeList=Array[FDfactor,nn,1,Union];
FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Table[Total[FDfactor[n]/.FDrules],{n,nn}]

A300660 Number of unlabeled rooted phylogenetic trees with n (leaf-) nodes such that for each inner node all children are either leaves or roots of distinct subtrees.

Original entry on oeis.org

0, 1, 1, 2, 3, 6, 13, 30, 72, 182, 467, 1222, 3245, 8722, 23663, 64758, 178459, 494922, 1380105, 3867414, 10884821, 30756410, 87215419, 248117618, 707952902, 2025479210, 5809424605, 16700811214, 48113496645, 138884979562, 401645917999, 1163530868090

Offset: 0

Alois P. Heinz, Jun 18 2018

From Gus Wiseman, Jul 31 2018 and Feb 06 2020: (Start)
a(n) is the number of lone-child-avoiding rooted identity trees whose leaves form an integer partition of n. For example, the following are the a(6) = 13 lone-child-avoiding rooted identity trees whose leaves form an integer partition of 6.
  6,
  (15),
  (24),
  (123), (1(23)), (2(13)), (3(12)),
  (1(14)),
  (1(1(13))),
  (12(12)), (1(2(12))), (2(1(12))),
  (1(1(1(12)))).
(End)

			:   a(3) = 2:        :   a(4) = 3:                      :
:      o       o     :        o         o        o      :
:     / \     /|\    :       / \       / \     /( )\    :
:    o   N   N N N   :      o   N     o   N   N N N N   :
:   ( )              :     / \       /|\                :
:   N N              :    o   N     N N N               :
:                    :   ( )                            :
:                    :   N N                            :
From _Gus Wiseman_, Feb 06 2020: (Start)
The a(2) = 1 through a(6) = 13 unlabeled rooted phylogenetic semi-identity trees:
  (oo) (ooo)     (oooo)         (ooooo)             (oooooo)
       ((o)(oo)) ((o)(ooo))     ((o)(oooo))         ((o)(ooooo))
                 ((o)((o)(oo))) ((oo)(ooo))         ((oo)(oooo))
                                ((o)((o)(ooo)))     ((o)(oo)(ooo))
                                ((oo)((o)(oo)))     (((o)(oo))(ooo))
                                ((o)((o)((o)(oo)))) ((o)((o)(oooo)))
                                                    ((o)((oo)(ooo)))
                                                    ((oo)((o)(ooo)))
                                                    ((o)(oo)((o)(oo)))
                                                    ((o)((o)((o)(ooo))))
                                                    ((o)((oo)((o)(oo))))
                                                    ((oo)((o)((o)(oo))))
                                                    ((o)((o)((o)((o)(oo)))))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..2079
Wikipedia, Phylogenetic tree
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000081, A004111, A141268, A289501, A301467.
Cf. A000669, A001678, A005804, A292504, A300660, A316653, A316654, A316656.
The locally disjoint case is A316694.
Cf. A276625, A306200, A319312, A331679, A331686, A331875.

b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(a(i), j), j=0..n/i)))
    end:
a:= n-> `if`(n=0, 0, 1+b(n, n-1)):
seq(a(n), n=0..30);

b[0, ] = 1; b[, _?NonPositive] = 0;
b[n_, i_] := b[n, i] = Sum[b[n-i*j, i-1]*Binomial[a[i], j], {j, 0, n/i}];
a[0] = 0; a[n_] := a[n] = 1 + b[n, n-1];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, May 03 2019, from Maple *)
ursit[n_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[ursit/@ptn]],UnsameQ@@#&],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}],n];
Table[Length[ursit[n]],{n,10}] (* Gus Wiseman, Feb 06 2020 *)

a(n) ~ c * d^n / n^(3/2), where d = 3.045141208159736483720243229947630323380565686... and c = 0.2004129296838557718008171812000512670126... - Vaclav Kotesovec, Aug 27 2018

A060356 Expansion of e.g.f.: -LambertW(-x/(1+x)).

Original entry on oeis.org

0, 1, 0, 3, 4, 65, 306, 4207, 38424, 573057, 7753510, 134046671, 2353898196, 47602871329, 1013794852266, 23751106404495, 590663769125296, 15806094859299329, 448284980183376078, 13515502344669830287

Offset: 0

Vladeta Jovovic, Apr 01 2001

Also the number of labeled lone-child-avoiding rooted trees with n nodes. A rooted tree is lone-child-avoiding if it has no unary branchings, meaning every non-leaf node covers at least two other nodes. The unlabeled version is A001678(n + 1). - Gus Wiseman, Jan 20 2020

			From _Gus Wiseman_, Dec 31 2019: (Start)
Non-isomorphic representatives of the a(7) = 4207 trees, written as root[branches], are:
  1[2,3[4,5[6,7]]]
  1[2,3[4,5,6,7]]
  1[2[3,4],5[6,7]]
  1[2,3,4[5,6,7]]
  1[2,3,4,5[6,7]]
  1[2,3,4,5,6,7]
(End)

Harry J. Smith, Table of n, a(n) for n = 0..100
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. A052871, A060313.
Cf. A008297.
Column k=0 of A231602.
The unlabeled version is A001678(n + 1).
The case where the root is fixed is A108919.
Unlabeled rooted trees are counted by A000081.
Lone-child-avoiding rooted trees with labeled leaves are A000311.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Singleton-reduced rooted trees are counted by A330951.
Cf. A000669, A004111, A005121, A048816, A292504, A316651, A316652, A318231, A318813, A330465, A330624.

List([0..20],n->Sum([1..n],k->(-1)^(n-k)*Factorial(n)/Factorial(k) *Binomial(n-1,k-1)*k^(k-1))); # Muniru A Asiru, Feb 19 2018

seq(coeff(series( -LambertW(-x/(1+x)), x, n+1), x, n)*n!, n = 0..20); # G. C. Greubel, Mar 16 2020

CoefficientList[Series[-LambertW[-x/(1+x)], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
a[n_]:=If[n==1,1,n*Sum[Times@@a/@Length/@stn,{stn,Select[sps[Range[n-1]],Length[#]>1&]}]];
Array[a,10] (* Gus Wiseman, Dec 31 2019 *)

{ for (n=0, 100, f=n!; a=sum(k=1, n, (-1)^(n - k)*f/k!*binomial(n - 1, k - 1)*k^(k - 1)); write("b060356.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 04 2009

my(x='x+O('x^20)); concat([0], Vec(serlaplace(-lambertw(-x/(1+x))))) \\ G. C. Greubel, Feb 19 2018

a(n) = Sum_{k=1..n} (-1)^(n-k)*n!/k!*binomial(n-1, k-1)*k^(k-1). a(n) = Sum_{k=0..n} Stirling1(n, k)*A058863(k). - Vladeta Jovovic, Sep 17 2003
a(n) ~ n^(n-1) * (1-exp(-1))^(n+1/2). - Vaclav Kotesovec, Nov 27 2012
a(n) = n * A108919(n). - Gus Wiseman, Dec 31 2019

A007562 Number of planted trees where non-root, non-leaf nodes an even distance from root are of degree 2.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 20, 36, 72, 137, 275, 541, 1098, 2208, 4521, 9240, 19084, 39451, 82113, 171240, 358794, 753460, 1587740, 3353192, 7100909, 15067924, 32044456, 68272854, 145730675, 311575140, 667221030, 1430892924, 3072925944, 6607832422, 14226665499

Offset: 1

N. J. A. Sloane

There is no planted tree on one node by definition.
Column k=2 of A144018. - Alois P. Heinz, Oct 17 2012
It appears that a(n) is also the number of locally non-intersecting unlabeled rooted trees with n nodes, where a tree is locally non-intersecting if the branches directly under of any non-leaf node have empty intersection. - Gus Wiseman, Aug 22 2018

			G.f. = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 10*x^7 + 20*x^8 + 36*x^9 + ...
From _Joerg Arndt_, Jun 23 2014: (Start)
The a(8) = 20 such trees have the following level sequences:
01:  [ 0 1 2 3 4 3 2 1 ]
02:  [ 0 1 2 3 3 3 2 1 ]
03:  [ 0 1 2 3 3 2 2 1 ]
04:  [ 0 1 2 3 3 2 1 1 ]
05:  [ 0 1 2 3 2 3 2 1 ]
06:  [ 0 1 2 3 2 2 2 1 ]
07:  [ 0 1 2 3 2 2 1 1 ]
08:  [ 0 1 2 3 2 1 2 1 ]
09:  [ 0 1 2 3 2 1 1 1 ]
10:  [ 0 1 2 2 2 2 2 1 ]
11:  [ 0 1 2 2 2 2 1 1 ]
12:  [ 0 1 2 2 2 1 2 1 ]
13:  [ 0 1 2 2 2 1 1 1 ]
14:  [ 0 1 2 2 1 2 2 1 ]
15:  [ 0 1 2 2 1 2 1 1 ]
16:  [ 0 1 2 2 1 1 1 1 ]
17:  [ 0 1 2 1 2 1 2 1 ]
18:  [ 0 1 2 1 2 1 1 1 ]
19:  [ 0 1 2 1 1 1 1 1 ]
20:  [ 0 1 1 1 1 1 1 1 ]
Successive levels change by at most 1 and the last level is 1, compare to the example in A000081.
(End)
From _Gus Wiseman_, Aug 22 2018: (Start)
The a(7) = 10 locally non-intersecting trees:
  (o(o(oo)))
  (o(oo(o)))
  (o(oooo))
  (oo(o(o)))
  (oo(ooo))
  (o(o)(oo))
  (ooo(oo))
  (oo(o)(o))
  (oooo(o))
  (oooooo)
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A000081, A000837, A004111, A007560, A144018, A289509, A316470, A316473, A316475.

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1 else (add(d*p(d), d=divisors(n)) +add(add(d*p(d), d= divisors(j)) *b(n-j), j=1..n-1))/n fi end end: b:= etr(a): a:= n-> `if`(n<=1, n, b(n-2)): seq(a(n), n=1..40);  # Alois P. Heinz, Sep 06 2008

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, (Sum[ Sum[ d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}] + Sum[ d*p[d], {d, Divisors[n]}])/n]; b]; b = etr[a]; a[n_] := If[n <= 1, n, b[n-2]]; Table[a[n], {n, 1, 36}] (* Jean-François Alcover, Aug 01 2013, after Alois P. Heinz *)
purt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]],Intersection@@#=={}&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[purt[n]],{n,10}] (* Gus Wiseman, Aug 22 2018 *)

{a(n) = local(A); if( n<2, n>0, A = x / (1 - x) + O(x^n); for(k=2, n-2, A /= (1 - x^k + O(x^n))^polcoeff(A, k-1)); polcoeff(A, n-1))}; /* Michael Somos, Oct 06 2003 */

Shifts left 2 places under Euler transform.
G.f.: x + x^2 / (Product_{k>0} (1 - x^k)^a(k)). - Michael Somos, Oct 06 2003
a(n) ~ c * d^n / n^(3/2), where d = 2.246066877341161662499621547921... and  c = 0.68490297576105466417608032... . - Vaclav Kotesovec, Jun 23 2014
G.f. A(x) satisfies: A(x) = x + x^2 * exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...). - Ilya Gutkovskiy, Jun 11 2021

Better description from Christian G. Bower, May 15 1998

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

A290760 Matula-Goebel numbers of transitive rooted identity trees (or transitive finitary sets).

Original entry on oeis.org

1, 2, 6, 30, 78, 330, 390, 870, 1410, 3198, 3390, 4290, 7878, 9570, 10230, 11310, 13026, 15510, 15990, 18330, 26070, 30966, 37290, 39390, 40890, 44070, 45210, 65130, 84810, 94830, 98310, 104610, 122070, 124410, 132990, 154830, 159330, 175890, 198330, 201630

Offset: 1

Gus Wiseman, Oct 19 2017

nonn

A rooted tree is transitive if every terminal subtree is a branch of the root. A finitary set is transitive if every element is also a subset.

			Let o = {}. The sequence of transitive finitary sets begins:
1     o
2     {o}
6     {o,{o}}
30    {o,{o},{{o}}}
78    {o,{o},{o,{o}}}
330   {o,{o},{{o}},{{{o}}}}
390   {o,{o},{{o}},{o,{o}}}
870   {o,{o},{{o}},{o,{{o}}}}
1410  {o,{o},{{o}},{{o},{{o}}}}
3198  {o,{o},{o,{o}},{{o,{o}}}}
3390  {o,{o},{{o}},{o,{o},{{o}}}}
4290  {o,{o},{{o}},{{{o}}},{o,{o}}}
7878  {o,{o},{o,{o}},{o,{o,{o}}}}
9570  {o,{o},{{o}},{{{o}}},{o,{{o}}}}
10230 {o,{o},{{o}},{{{o}}},{{{{o}}}}}
11310 {o,{o},{{o}},{o,{o}},{o,{{o}}}}
13026 {o,{o},{o,{o}},{{o},{o,{o}}}}
15510 {o,{o},{{o}},{{{o}}},{{o},{{o}}}}
15990 {o,{o},{{o}},{o,{o}},{{o,{o}}}}
18330 {o,{o},{{o}},{o,{o}},{{o},{{o}}}}

Cf. A000081, A001192, A004111, A007097, A076146, A276625, A279861, A290689, A290822.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
finitaryQ[n_]:=finitaryQ[n]=Or[n===1,With[{m=primeMS[n]},{UnsameQ@@m,finitaryQ/@m}]/.List->And];
subprimes[n_]:=If[n===1,{},Union@@Cases[FactorInteger[n],{p_,_}:>FactorInteger[PrimePi[p]][[All,1]]]];
transitaryQ[n_]:=Divisible[n,Times@@subprimes[n]];
nn=100000;Fold[Select,Range[nn],{finitaryQ,transitaryQ}]

cp's OEIS Frontend

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

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A298422 Number of rooted trees with n nodes in which all positive outdegrees are the same.

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A301700 Number of aperiodic rooted trees with n nodes.

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A111299 Numbers whose Matula tree is a binary tree (i.e., root has degree 2 and all nodes except root and leaves have degree 3).

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A299757 Weight of the strict integer partition with FDH number n.

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A300660 Number of unlabeled rooted phylogenetic trees with n (leaf-) nodes such that for each inner node all children are either leaves or roots of distinct subtrees.

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A060356 Expansion of e.g.f.: -LambertW(-x/(1+x)).

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