A001678 -id:A001678 - cp's OEIS frontend

A331965 Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees.

1, 4, 8, 14, 16, 28, 32, 38, 56, 64, 76, 86, 106, 112, 128, 133, 152, 172, 212, 214, 224, 256, 262, 266, 301, 304, 326, 344, 371, 424, 428, 448, 512, 524, 526, 532, 602, 608, 622, 652, 688, 742, 749, 766, 817, 848, 856, 886, 896, 917, 1007, 1024, 1048, 1052

Offset: 1

Gus Wiseman, Feb 04 2020

nonn

First differs from A331683 in having 133, the Matula-Goebel number of the tree ((oo)(ooo)).
Lone-child-avoiding means there are no unary branchings.
In a semi-identity tree, the non-leaf branches of any given vertex are all distinct.
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 composite numbers that are n times a power of two, where n is a squarefree number whose prime indices already belong to the sequence, and a prime index of n is a number m such that prime(m) divides n. [Clarified by Peter Munn and Gus Wiseman, Jun 24 2021]

			The sequence of all lone-child-avoiding rooted semi-identity 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))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  133: ((oo)(ooo))
  152: (ooo(ooo))
  172: (oo(o(oo)))
  212: (oo(oooo))
  214: (o(oo(oo)))
The sequence of terms together with their prime indices begins:
    1: {}                 224: {1,1,1,1,1,4}
    4: {1,1}              256: {1,1,1,1,1,1,1,1}
    8: {1,1,1}            262: {1,32}
   14: {1,4}              266: {1,4,8}
   16: {1,1,1,1}          301: {4,14}
   28: {1,1,4}            304: {1,1,1,1,8}
   32: {1,1,1,1,1}        326: {1,38}
   38: {1,8}              344: {1,1,1,14}
   56: {1,1,1,4}          371: {4,16}
   64: {1,1,1,1,1,1}      424: {1,1,1,16}
   76: {1,1,8}            428: {1,1,28}
   86: {1,14}             448: {1,1,1,1,1,1,4}
  106: {1,16}             512: {1,1,1,1,1,1,1,1,1}
  112: {1,1,1,1,4}        524: {1,1,32}
  128: {1,1,1,1,1,1,1}    526: {1,56}
  133: {4,8}              532: {1,1,4,8}
  152: {1,1,1,8}          602: {1,4,14}
  172: {1,1,14}           608: {1,1,1,1,1,8}
  212: {1,1,16}           622: {1,64}
  214: {1,28}             652: {1,1,38}

The non-semi case is {1}.
Not requiring lone-child-avoidance gives A306202.
The locally disjoint version is A331683.
These trees are counted by A331966.
The semi-lone-child-avoiding case is A331994.
Matula-Goebel numbers of rooted identity trees are A276625.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Semi-identity trees are counted by A306200.
Cf. A001678, A004111, A007097, A061775, A122132, A196050, A300660, A316694, A320269, A331681, A331686, A331875, A331936, A331937, A331993.

csiQ[n_]:=n==1||!PrimeQ[n]&&FreeQ[FactorInteger[n],{?(#>2&),?(#>1&)}]&&And@@csiQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],csiQ]

Intersection of A291636 and A306202.

A331966 Number of lone-child-avoiding rooted semi-identity trees with n vertices.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 9, 16, 30, 55, 105, 200, 388, 754, 1483, 2923, 5807, 11575, 23190, 46608, 94043, 190287, 386214, 785831, 1602952, 3276845, 6712905, 13778079, 28330583, 58350582, 120370731, 248676129, 514459237, 1065696295, 2210302177, 4589599429, 9540623926

Offset: 1

Gus Wiseman, Feb 05 2020

nonn

Lone-child-avoiding means there are no unary branchings.

In a semi-identity tree, the non-leaf branches of any given vertex are distinct.

			The a(1) = 1 through a(9) = 16 trees (empty column shown as dot):
  o  .  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)    (oooooooo)
                     (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))   (o(oooooo))
                              (oo(oo))  (oo(ooo))   (oo(oooo))   (oo(ooooo))
                                        (ooo(oo))   (ooo(ooo))   (ooo(oooo))
                                        (o(o(oo)))  (oooo(oo))   (oooo(ooo))
                                                    ((oo)(ooo))  (ooooo(oo))
                                                    (o(o(ooo)))  ((oo)(oooo))
                                                    (o(oo(oo)))  (o(o(oooo)))
                                                    (oo(o(oo)))  (o(oo)(ooo))
                                                                 (o(oo(ooo)))
                                                                 (o(ooo(oo)))
                                                                 (oo(o(ooo)))
                                                                 (oo(oo(oo)))
                                                                 (ooo(o(oo)))
                                                                 ((oo)(o(oo)))
                                                                 (o(o(o(oo))))

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

The non-semi case is A000007.
Lone-child-avoiding rooted trees are A001678.
The locally disjoint case is A212804.
Not requiring lone-child-avoidance gives A306200.
Matula-Goebel numbers of these trees are A331965.
The semi-lone-child-avoiding version is A331993.
Cf. A000081, A004111, A291636, A300660, A306202, A316694, A331683, A331686, A331783, A331875, A331964, A331994.

ssb[n_]:=If[n==1,{{}},Join@@Function[c,Select[Union[Sort/@Tuples[ssb/@c]],UnsameQ@@DeleteCases[#,{}]&]]/@Rest[IntegerPartitions[n-1]]];
Table[Length[ssb[n]],{n,10}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=[0, 0]); for(n=2, n-1, v=concat(v, 1 + vecsum(WeighT(v)) - v[n])); v[1]=1; v} \\ Andrew Howroyd, Feb 09 2020

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

A198518 G.f. satisfies: A(x) = exp( Sum_{n>=1} A(x^n)/(1+x^n) * x^n/n ).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 16, 29, 54, 102, 194, 375, 730, 1434, 2837, 5650, 11311, 22767, 46023, 93422, 190322, 389037, 797613, 1639878, 3380099, 6983484, 14459570, 29999618, 62357426, 129843590, 270807835, 565674584, 1183301266, 2478624060, 5198504694, 10916110768, 22948299899

Offset: 0

Paul D. Hanna, Oct 26 2011

nonn

For n>=1, a(n) is the number of rooted trees (see A000081) with n non-root nodes where non-root nodes cannot have out-degree 1, see the note by David Callan and the example. Imposing the condition also for the root node gives A001678. - Joerg Arndt, Jun 28 2014
Compare definition to G(x) = exp( Sum_{n>=1} G(x^n)*x^n/n ), where G(x) is the g.f. of A000081, the number of rooted trees with n nodes.
Number of forests of lone-child-avoiding rooted trees with n unlabeled vertices. - Gus Wiseman, Feb 03 2020

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 9*x^6 + 16*x^7 + 29*x^8 +...
where
log(A(x)) = A(x)/(1+x)*x + A(x^2)/(1+x^2)*x^2/2 + A(x^3)/(1+x^3)*x^3/3 +...
The coefficients in A(x)/(1+x) begin:
[1, 0, 1, 1, 2, 3, 6, 10, 19, 35, 67, 127, 248, 482, 952, 1885, 3765, ...]
(this is, up to offset, A001678),
from which g.f. A(x) may be generated by the Euler transform:
A(x) = 1/((1-x)^1*(1-x^2)^0*(1-x^3)^1*(1-x^4)^1*(1-x^5)^2*(1-x^6)^3*(1-x^7)^6*(1-x^8)^10*(1-x^9)^19*(1-x^10)^35*...).
From _Joerg Arndt_, Jun 28 2014: (Start)
The a(6) = 9 rooted trees with 6 non-root nodes as described in the comment are:
:           level sequence       out-degrees (dots for zeros)
:     1:  [ 0 1 2 3 3 3 2 ]    [ 1 2 3 . . . . ]
:  O--o--o--o
:        .--o
:        .--o
:     .--o
:
:     2:  [ 0 1 2 3 3 2 2 ]    [ 1 3 2 . . . . ]
:  O--o--o--o
:        .--o
:     .--o
:     .--o
:
:     3:  [ 0 1 2 3 3 2 1 ]    [ 2 2 2 . . . . ]
:  O--o--o--o
:        .--o
:     .--o
:  .--o
:
:     4:  [ 0 1 2 2 2 2 2 ]    [ 1 5 . . . . . ]
:  O--o--o
:     .--o
:     .--o
:     .--o
:     .--o
:
:     5:  [ 0 1 2 2 2 2 1 ]    [ 2 4 . . . . . ]
:  O--o--o
:     .--o
:     .--o
:     .--o
:  .--o
:
:     6:  [ 0 1 2 2 2 1 1 ]    [ 3 3 . . . . . ]
:  O--o--o
:     .--o
:     .--o
:  .--o
:  .--o
:
:     7:  [ 0 1 2 2 1 2 2 ]    [ 2 2 . . 2 . . ]
:  O--o--o
:     .--o
:  .--o--o
:     .--o
:
:     8:  [ 0 1 2 2 1 1 1 ]    [ 4 2 . . . . . ]
:  O--o--o
:     .--o
:  .--o
:  .--o
:  .--o
:
:     9:  [ 0 1 1 1 1 1 1 ]    [ 6 . . . . . . ]
:  O--o
:  .--o
:  .--o
:  .--o
:  .--o
:  .--o
(End)
From _Gus Wiseman_, Jan 22 2020: (Start)
The a(0) = 1 through a(6) = 9 rooted trees with n + 1 nodes where non-root vertices cannot have out-degree 1:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)    (oooooo)
                ((oo))  ((ooo))  ((oooo))   ((ooooo))
                        (o(oo))  (o(ooo))   (o(oooo))
                                 (oo(oo))   (oo(ooo))
                                 ((o(oo)))  (ooo(oo))
                                            ((o(ooo)))
                                            ((oo)(oo))
                                            ((oo(oo)))
                                            (o(o(oo)))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
David Callan, Rooted trees with no out-degree = 1, (7-July-2014).
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. A052855, A246403.
The labeled version is A254382.
Unlabeled rooted trees are A000081.
Lone-child-avoiding rooted trees are A001678(n+1).
Topologically series-reduced rooted trees are A001679.
Labeled lone-child-avoiding rooted trees are A060356.
Cf. A000669, A004111, A108919, A291636, A330951, A331488, A331934.

with(numtheory):
b:= proc(n) b(n):= `if`(n=0, 1, a(n)-b(n-1)) end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(
       d*b(d-1), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 02 2014

b[n_] := b[n] = If[n==0, 1, a[n] - b[n-1]];
a[n_] := a[n] = If[n==0, 1, Sum[Sum[d*b[d-1], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 21 2017, after Alois P. Heinz *)
urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[Length[Select[urt[n],FreeQ[Z@@#,{}]&]],{n,10}] (* _Gus Wiseman, Jan 22 2020 *)

{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,subst(A/(1+x),x,x^m+x*O(x^n))*x^m/m)));polcoeff(A,n)}

Euler transform of coefficients in A(x)/(1+x), where g.f. A(x) = Sum_{n>=0} a(n)*x^n.
a(n) ~ c * d^n / n^(3/2), where d = A246403 = 2.18946198566085056388702757711..., c = 1.3437262442171062526771597... . - Vaclav Kotesovec, Sep 03 2014
a(n) = A001678(n + 1) + A001678(n + 2). - Gus Wiseman, Jan 22 2020
Euler transform of A001678(n + 1). - Gus Wiseman, Feb 03 2020

A244454 Number T(n,k) of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 7, 1, 0, 1, 0, 17, 2, 0, 0, 1, 0, 42, 4, 1, 0, 0, 1, 0, 105, 7, 2, 0, 0, 0, 1, 0, 267, 15, 2, 1, 0, 0, 0, 1, 0, 684, 28, 4, 2, 0, 0, 0, 0, 1, 0, 1775, 56, 7, 2, 1, 0, 0, 0, 0, 1, 0, 4639, 110, 12, 2, 2, 0, 0, 0, 0, 0, 1

Offset: 1

Joerg Arndt and Alois P. Heinz, Jun 28 2014

T(1,0) = 1 by convention.

Sum_{i=2..n-1} T(n,i) = A001678(n+1) for n>1.

			The A000081(5) = 9 rooted trees with 5 nodes sorted by minimal outdegree of inner nodes 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                                     :         :         :
:                                       :         :         :
: ------------------1------------------ : ---2--- : ---4--- :
Thus row 5 = [0, 7, 1, 0, 1].
Triangle T(n,k) begins:
  1;
  0,    1;
  0,    1,   1;
  0,    3,   0,  1;
  0,    7,   1,  0, 1;
  0,   17,   2,  0, 0, 1;
  0,   42,   4,  1, 0, 0, 1;
  0,  105,   7,  2, 0, 0, 0, 1;
  0,  267,  15,  2, 1, 0, 0, 0, 1;
  0,  684,  28,  4, 2, 0, 0, 0, 0, 1;
  0, 1775,  56,  7, 2, 1, 0, 0, 0, 0, 1;
  0, 4639, 110, 12, 2, 2, 0, 0, 0, 0, 0, 1;

Alois P. Heinz, Rows n = 1..141, flattened

Columns k=0-10 give: A063524, A244455, A244456, A244457, A244458, A244459, A244460, A244461, A244462, A244463, A244464.
Row sums give A000081.
Cf. A001678, A244372, A244530 (ordered unlabeled rooted trees).

b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],
      1, 0), `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
      b(n-i*j, i-1, max(0, t-j), k), j=0..n/i)))
    end:
T:= (n, k)-> b(n-1$2, k$2) -`if`(n=1 and k=0, 0, b(n-1$2, k+1$2)):
seq(seq(T(n, k), k=0..n-1), n=1..14);

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[i<1, 0, Sum[Binomial[b[i-1, i-1, k, k]+j-1, j]* b[n-i*j, i-1, Max[0, t-j], k], {j, 0, n/i}]]]; T[n_, k_] := b[n-1, n-1, k, k] - If[n == 1 && k == 0, 0, b[n-1, n-1, k+1, k+1]]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 08 2015, translated from Maple *)

A316696 Number of lone-child-avoiding locally disjoint rooted trees whose leaves form an integer partition of n.

Original entry on oeis.org

1, 2, 4, 11, 27, 80, 218, 654, 1923, 5924, 18310, 58176, 186341, 606814, 1993420, 6618160, 22134640

Offset: 1

Gus Wiseman, Jul 10 2018

A rooted tree is lone-child-avoiding if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other (unequal) branch of the same root.

			The a(4) = 11 rooted trees:
  4,
  (13),
  (22),
  (1(12)), (2(11)), (112),
  (1(1(11))), (1(111)), ((11)(11)), (11(11)), (1111).

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Matula-Goebel numbers of locally disjoint rooted trees are A316495.
The case where all leaves are 1's is A316697.
Lone-child-avoiding locally disjoint rooted trees are A331680.
Cf. A000669, A001678, A141268, A316473, A316652, A331678, A331686, A331687, A331871, A331872, A331874.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],disjointQ],{ptn,Rest[IntegerPartitions[n]]}],{n}];
Table[Length[nms[n]],{n,10}]

a(16)-a(17) from Robert Price, Sep 16 2018

Terminology corrected by Gus Wiseman, Feb 06 2020

A317875 Number of achiral free pure multifunctions with n unlabeled leaves.

Original entry on oeis.org

1, 1, 3, 9, 30, 102, 369, 1362, 5181, 20064, 79035, 315366, 1272789, 5185080, 21296196, 88083993, 366584253, 1533953100, 6449904138, 27238006971, 115475933202, 491293053093, 2096930378415, 8976370298886, 38528771056425, 165784567505325

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

An achiral free pure multifunction is either (case 1) the leaf symbol "o", or (case 2) a nonempty expression of the form h[g, ..., g], where h and g are both achiral free pure multifunctions.

			The first 4 terms count the following multifunctions.
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].

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A277996, A280000, A317853.
Cf. A317876, A317877, A317878, A317879, A317880, A317881.
Cf. A317882, A317883, A317884, A317885.

a[n_]:=If[n==1,1,Sum[a[n-k]*Sum[a[d],{d,Divisors[k]}],{k,n-1}]];
Array[a,12]

seq(n)={my(p=O(x)); for(n=1, n, p = x + p*(sum(k=1, n-1, subst(p + O(x^(n\k+1)), x, x^k)) ) + O(x*x^n)); Vec(p)} \\ Andrew Howroyd, Aug 19 2018

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=sum(i=1, n-1, v[i]*sumdiv(n-i, d, v[d]))); v} \\ Andrew Howroyd, Aug 19 2018

a(1) = 1; a(n > 1) = Sum_{0 < k < n} a(n - k) * Sum_{d|k} a(d).
From Ilya Gutkovskiy, Apr 30 2019: (Start)
G.f. A(x) satisfies: A(x) = x + A(x) * Sum_{k>=1} A(x^k).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x + (Sum_{n>=1} a(n)*x^n) * (Sum_{n>=1} a(n)*x^n/(1 - x^n)). (End)

A330475 Number of balanced reduced multisystems whose atoms constitute a strongly normal multiset of size n.

Original entry on oeis.org

1, 1, 2, 9, 85, 1143, 25270

Offset: 0

Gus Wiseman, Dec 27 2019

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

A finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.

			The a(0) = 1 through a(3) = 9 multisystems:
  {}  {1}  {1,1}  {1,1,1}
           {1,2}  {1,1,2}
                  {1,2,3}
                  {{1},{1,1}}
                  {{1},{1,2}}
                  {{1},{2,3}}
                  {{2},{1,1}}
                  {{2},{1,3}}
                  {{3},{1,2}}

The (weakly) normal version is A330655.
The maximum-depth case is A330675.
The case where the atoms are {1..n} is A005121.
The case where the atoms are all 1's is A318813.
The tree version is A330471.
Multiset partitions of strongly normal multisets are A035310.
Cf. A000311, A000669, A001678, A316652, A318812, A330467, A330474, A330628, A330663, A330679.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

A331936 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex (semi-achirality).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 14, 16, 18, 24, 26, 27, 28, 32, 36, 38, 46, 48, 49, 52, 54, 56, 64, 72, 74, 76, 81, 86, 92, 96, 98, 104, 106, 108, 112, 122, 128, 144, 148, 152, 162, 169, 172, 178, 184, 192, 196, 202, 206, 208, 212, 214, 216, 224, 243, 244, 256, 262, 288

Offset: 1

Gus Wiseman, Feb 03 2020

nonn

First differs from A331873 in lacking 69, the Matula-Goebel number of the tree ((o)((o)(o))).
A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf.
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 1, 2, and all numbers equal to a power of 2 (other than 1) times a power of prime(j) for some j > 1 already in the sequence.

			The sequence of rooted trees ranked by this sequence together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
   9: ((o)(o))
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  18: (o(o)(o))
  24: (ooo(o))
  26: (o(o(o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  46: (o((o)(o)))
  48: (oooo(o))
  49: ((oo)(oo))
The sequence of terms together with their prime indices begins:
    1: {}              52: {1,1,6}            152: {1,1,1,8}
    2: {1}             54: {1,2,2,2}          162: {1,2,2,2,2}
    4: {1,1}           56: {1,1,1,4}          169: {6,6}
    6: {1,2}           64: {1,1,1,1,1,1}      172: {1,1,14}
    8: {1,1,1}         72: {1,1,1,2,2}        178: {1,24}
    9: {2,2}           74: {1,12}             184: {1,1,1,9}
   12: {1,1,2}         76: {1,1,8}            192: {1,1,1,1,1,1,2}
   14: {1,4}           81: {2,2,2,2}          196: {1,1,4,4}
   16: {1,1,1,1}       86: {1,14}             202: {1,26}
   18: {1,2,2}         92: {1,1,9}            206: {1,27}
   24: {1,1,1,2}       96: {1,1,1,1,1,2}      208: {1,1,1,1,6}
   26: {1,6}           98: {1,4,4}            212: {1,1,16}
   27: {2,2,2}        104: {1,1,1,6}          214: {1,28}
   28: {1,1,4}        106: {1,16}             216: {1,1,1,2,2,2}
   32: {1,1,1,1,1}    108: {1,1,2,2,2}        224: {1,1,1,1,1,4}
   36: {1,1,2,2}      112: {1,1,1,1,4}        243: {2,2,2,2,2}
   38: {1,8}          122: {1,18}             244: {1,1,18}
   46: {1,9}          128: {1,1,1,1,1,1,1}    256: {1,1,1,1,1,1,1,1}
   48: {1,1,1,1,2}    144: {1,1,1,1,2,2}      262: {1,32}
   49: {4,4}          148: {1,1,12}           288: {1,1,1,1,1,2,2}

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.

A superset of A000079.
The non-lone-child-avoiding version is A320230.
The non-semi version is A320269.
These trees are counted by A331933.
Not requiring semi-achirality gives A331935.
The fully-achiral case is A331992.
Achiral trees are counted by A003238.
Numbers with at most one distinct odd prime factor are A070776.
Matula-Goebel numbers of achiral rooted trees are A214577.
Matula-Goebel numbers of semi-identity trees are A306202.
Numbers S with at most one distinct prime index in S are A331912.
Cf. A001678, A007097, A050381, A061775, A196050, A291636, A331784, A331873, A331914, A331934, A331965, A331967, A331991.

msQ[n_]:=n<=2||!PrimeQ[n]&&Length[DeleteCases[FactorInteger[n],{2,_}]]<=1&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],msQ]

Intersection of A320230 and A331935.

A060313 Number of homeomorphically irreducible rooted trees (also known as series-reduced rooted trees, or rooted trees without nodes of degree 2) on n labeled nodes.

Original entry on oeis.org

1, 2, 0, 16, 25, 576, 2989, 51584, 512649, 8927200, 130956001, 2533847328, 48008533885, 1059817074512, 24196291364925, 609350187214336, 16135860325700881, 459434230368302016, 13788624945433889593, 439102289933675933600, 14705223056221892676741

Offset: 1

Vladeta Jovovic, Mar 27 2001

			From _Gus Wiseman_, Jan 22 2020: (Start)
The a(1) = 1 through a(4) = 16 trees (in the format root[branches], empty column shown as dot) are:
  1  1[2]  .  1[2,3,4]
     2[1]     1[2[3,4]]
              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)

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

G. C. Greubel, Table of n, a(n) for n = 1..400
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 unlabeled unrooted version is A000014.
The unrooted version is A005512.
The unlabeled version is A001679 or A059123.
The lone-child-avoiding version is A060356.
Labeled rooted trees are A000169.
Cf. A000669, A001678, A108919, A254382, A331489, A331578.

[1] cat [n*Factorial(n-2)*(&+[(-1)^k*Binomial(n,k)*(n-k)^(n-k-2)/Factorial(n-k-2): k in [0..n-2]]): n in [2..20]]; // G. C. Greubel, Mar 07 2020

seq( `if`(n=1, 1, n*(n-2)!*add((-1)^k*binomial(n, k)*(n-k)^(n-k-2)/(n-k-2)!, k=0..n-2)), n=1..20); # G. C. Greubel, Mar 07 2020

f[n_] := If[n < 2, 1, n(n - 2)!Sum[(-1)^k*Binomial[n, k](n - k)^(n - 2 - k)/(n - 2 - k)!, {k, 0, n - 2}]]; Table[ f[n], {n, 19}] (* Robert G. Wilson v, Feb 12 2005 *)
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]],Length[#]!=2&&FreeQ[Z@@#,Integer[]]&]],{n,6}] (* Gus Wiseman, Jan 22 2020 *)

[1]+[n*factorial(n-2)*sum((-1)^k*binomial(n,k)*(n-k)^(n-k-2)/factorial( n-k-2) for k in (0..n-2)) for n in (2..20)] # G. C. Greubel, Mar 07 2020

a(n) = n*(n-2)!*Sum_{k=0..n-2} (-1)^k*binomial(n, k)*(n-k)^(n-k-2)/(n-k-2)!, n>1.
E.g.f.: x*(exp( - LambertW(-x/(1+x))) - (LambertW(-x/(1+x))/2 )^2).
a(n) ~ n^(n-1) * (1-exp(-1))^(n+1/2). - Vaclav Kotesovec, Oct 05 2013
E.g.f.: -(1+x)*LambertW(-x/(1+x)) - (x/2)*LambertW(-x/(1+x))^2. - G. C. Greubel, Mar 07 2020

A316624 Number of balanced p-trees with n leaves.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 8, 9, 16, 20, 40, 47, 83, 111, 201, 259, 454, 603, 1049, 1432, 2407, 3390, 6006, 8222, 13904, 20304, 34828, 50291, 85817, 126013, 217653, 317894, 535103, 798184, 1367585, 2008125, 3360067, 5048274, 8499942, 12623978, 21023718, 31552560, 52575257

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

A p-tree of weight n is either a single node (if n = 1) or a finite sequence of p-trees whose weights are weakly decreasing and sum to n.

A tree is balanced if all leaves have the same height.

			The a(1) = 1 through a(7) = 4 balanced p-trees:
  o  (oo)  (ooo)  (oooo)      (ooooo)      (oooooo)        (ooooooo)
                  ((oo)(oo))  ((ooo)(oo))  ((ooo)(ooo))    ((oooo)(ooo))
                                           ((oooo)(oo))    ((ooooo)(oo))
                                           ((oo)(oo)(oo))  ((ooo)(oo)(oo))

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

Cf. A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A196545, A289501, A319312.

ptrs[n_]:=If[n==1,{"o"},Join@@Table[Tuples[ptrs/@p],{p,Rest[IntegerPartitions[n]]}]];
Table[Length[ptrs[n]],{n,12}]
Table[Length[Select[ptrs[n],SameQ@@Length/@Position[#,"o"]&]],{n,12}]

seq(n)={my(p=x + O(x*x^n), q=0); while(p, q+=p; p = 1/prod(k=1, n, 1 - polcoef(p,k)*x^k + O(x*x^n)) - 1 - p); Vec(q)} \\ Andrew Howroyd, Oct 26 2018

Terms a(17) and beyond from Andrew Howroyd, Oct 26 2018

cp's OEIS Frontend

A331965 Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees.

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A331966 Number of lone-child-avoiding rooted semi-identity trees with n vertices.

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A198518 G.f. satisfies: A(x) = exp( Sum_{n>=1} A(x^n)/(1+x^n) * x^n/n ).

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A244454 Number T(n,k) of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

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A316696 Number of lone-child-avoiding locally disjoint rooted trees whose leaves form an integer partition of n.

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A317875 Number of achiral free pure multifunctions with n unlabeled leaves.

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A330475 Number of balanced reduced multisystems whose atoms constitute a strongly normal multiset of size n.

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