A330951 -id:A330951 - cp's OEIS frontend

A001678 Number of series-reduced planted trees with n nodes.

0, 0, 1, 0, 1, 1, 2, 3, 6, 10, 19, 35, 67, 127, 248, 482, 952, 1885, 3765, 7546, 15221, 30802, 62620, 127702, 261335, 536278, 1103600, 2276499, 4706985, 9752585, 20247033, 42110393, 87733197, 183074638, 382599946, 800701320, 1677922740, 3520581954

Offset: 0

N. J. A. Sloane

The initial term is 0 by convention, though a good case can be made that it should be 1 instead.
Series-reduced trees contain no node with valency 2; see A000014 for the unrooted series-reduced trees. - Joerg Arndt, Mar 03 2015
For n>=2, a(n+1) is the number of unordered rooted trees (see A000081) with n nodes where nodes cannot have out-degree 1, see example. Imposing the condition only at non-root nodes gives A198518. - Joerg Arndt, Jun 28 2014
For n>=3, a(n+1) is the number of unordered rooted trees with n nodes where all limbs are of length >= 2. Limbs are the paths from the leafs (towards the root) to the nearest branching point (with the root considered to be a branching point). - Joerg Arndt, Mar 03 2015
A rooted tree is lone-child-avoiding if no vertex has exactly one child, and topologically series-reduced if no vertex has degree 2. This sequence counts unlabeled lone-child-avoiding rooted trees with n - 1 vertices. Topologically series-reduced rooted trees are counted by A001679, which is essentially the same as A059123. - Gus Wiseman, Jan 20 2020

			--------------- Examples (i=internal,e=external): ---------------------------
|.n=2.|..n=4..|..n=5..|...n=6.............|....n=7..........................|
|.....|.......|.......|.............e...e.|................e.e.e......e...e.|
|.....|.e...e.|.e.e.e.|.e.e.e.e...e...i...|.e.e.e.e.e...e....i....e.e...i...|
|..e..|...i...|...i...|....i........i.....|.....i..........i..........i.....|
|..e..|...e...|...e...|....e........e.....|.....e..........e..........e.....|
-----------------------------------------------------------------------------
G.f. = x^2 + x^4 + x^5 + 2*x^6 + 3*x^7 + 6*x^8 + 10*x^9 + 19*x^10 + ...
From _Joerg Arndt_, Jun 28 2014: (Start)
The a(8) = 6 rooted trees with 7 nodes as described in the comment are:
:           level sequence       out-degrees (dots for zeros)
:     1:  [ 0 1 2 3 3 2 1 ]    [ 2 2 2 . . . . ]
:  O--o--o--o
:        .--o
:     .--o
:  .--o
:
:     2:  [ 0 1 2 2 2 2 1 ]    [ 2 4 . . . . . ]
:  O--o--o
:     .--o
:     .--o
:     .--o
:  .--o
:
:     3:  [ 0 1 2 2 2 1 1 ]    [ 3 3 . . . . . ]
:  O--o--o
:     .--o
:     .--o
:  .--o
:  .--o
:
:     4:  [ 0 1 2 2 1 2 2 ]    [ 2 2 . . 2 . . ]
:  O--o--o
:     .--o
:  .--o--o
:     .--o
:
:     5:  [ 0 1 2 2 1 1 1 ]    [ 4 2 . . . . . ]
:  O--o--o
:     .--o
:  .--o
:  .--o
:  .--o
:
:     6:  [ 0 1 1 1 1 1 1 ]    [ 6 . . . . . . ]
:  O--o
:  .--o
:  .--o
:  .--o
:  .--o
:  .--o
:
(End)
From _Gus Wiseman_, Jan 20 2020: (Start)
The a(2) = 1 through a(9) = 10 unlabeled lone-child-avoiding rooted trees with n - 1 nodes (empty n = 3 column shown as dot) are:
  o   .   (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)
                       (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))
                                (oo(oo))  (oo(ooo))   (oo(oooo))
                                          (ooo(oo))   (ooo(ooo))
                                          ((oo)(oo))  (oooo(oo))
                                          (o(o(oo)))  ((oo)(ooo))
                                                      (o(o(ooo)))
                                                      (o(oo)(oo))
                                                      (o(oo(oo)))
                                                      (oo(o(oo)))
(End)

D. G. Cantor, personal communication.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 525.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 62.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 = 0..1000 (first 501 terms from Christian G. Bower)
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
F. Harary and E. M. Palmer, Probability that a point of a tree is fixed, Math. Proc. Camb. Phil. Soc. 85 (1979) 407-415.
F. Harary and G. Prins, The number of homeomorphically irreducible trees and other species, Acta Math., 101 (1959), 141-162.
F. Harary, R. W. Robinson and A. J. Schwenk, Twenty-step algorithm for determining the asymptotic number of trees of various species, J. Austral. Math. Soc., Series A, 20 (1975), 483-503.
F. Harary, R. W. Robinson and A. J. Schwenk, Corrigenda: Twenty-step algorithm for determining the asymptotic number of trees of various species, J. Austral. Math. Soc., Series A 41 (1986), p. 325.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 404
Marko Riedel, Generating functions of unordered rooted trees with n nodes where nodes cannot have out-degree 1, classified by the number of leaves, using the Polya Enumeration Theorem and the exponential formula.
Eric Weisstein's World of Mathematics, Series-reduced 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. A000014, A007827, A246403, A005202.
Unlabeled rooted trees are counted by A000081.
Topologically series-reduced rooted trees are counted by A001679.
Labeled lone-child-avoiding rooted trees are counted by A060356.
Labeled lone-child-avoiding unrooted trees are counted by A108919.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Singleton-reduced rooted trees are counted by A330951.
Cf. A000669, A004111, A106179, A198518, A254382, A331488, A331578.

with (powseries): with (combstruct): n := 30: sys := {B = Prod(C,Z), S = Set(B,1 <= card), C = Union(Z,S)}: A001678 := 1,0,1,seq(count([S, sys, unlabeled],size=i),i=1..n); # Ulrich Schimke (ulrschimke(AT)aol.com)
# second Maple program:
with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
       d*a(d+1), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n<2, 0,
      `if`(n=2, 1, b(n-2)-a(n-1)))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 02 2014

b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*a[d+1], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; a[n_] := a[n] = If[n < 2, 0, If[n == 2, 1, b[n-2] - a[n-1]]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Sep 24 2014, after Alois P. Heinz *)
terms = 38; A[] = 0; Do[A[x] = (x^2/(1+x))*Exp[Sum[A[x^k]/(k*x^k), {k, 1, j}]] + O[x]^j // Normal, {j, 1, terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 12 2018 *)
urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[If[n<=1,0,Length[Select[urt[n-1],FreeQ[#,{}]&]]],{n,0,10}] (* _Gus Wiseman, Jan 20 2020 *)

(a(n) = if( n<4, n==2, T(n-2, n-3))); /* where */ {T(n, k) = if( n<1 || k<1, (n==0) && (k>=0), sum(j=1, k, sum(i=1, n\j, T(n-i*j, min(n-i*j, j-1)) * binomial( a(j+1) + i-1, i))))}; /* Michael Somos, Jun 04 2002 */

{a(n) = local(A); if( n<3, n==2, A = x / (1 - x^2) + O(x^n); for(k=3, n-2, A /= (1 - x^k + O(x^n))^polcoeff(A, k)); polcoeff(A, n-1))}; /* Michael Somos, Oct 06 2003 */

G.f.: A(x) satisfies A(x) = (x^2/(1+x))*exp( Sum_{k>=1} A(x^k)/(k*x^k) ) [Harary and E. M. Palmer, 1973, p. 62, Eq. (3.3.8)].
G.f.: A(x) = Sum_{n>=2} a(n) * x^n = x^2 / ((1 + x) * Product_{k>0} (1 - x^k)^a(k+1)). - Michael Somos, Oct 06 2003
a(n) ~ c * d^n / n^(3/2), where d = A246403 = 2.189461985660850563... and c = 0.1924225474701550354144525345664845514828912790855223729854471406053655209... - Vaclav Kotesovec, Jun 26 2014
a(n) = Sum_{i=2..n-2} A106179(i, n-1-i) for n >= 3. - Andrew Howroyd, Mar 29 2021

Additional comments from Michael Somos, Jun 05 2002

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

A331934 Number of semi-lone-child-avoiding rooted trees with n unlabeled vertices.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 15, 29, 62, 129, 279, 602, 1326, 2928, 6544, 14692, 33233, 75512, 172506, 395633, 911108, 2105261, 4880535, 11346694, 26451357, 61813588, 144781303, 339820852, 799168292, 1882845298, 4443543279, 10503486112, 24864797324, 58944602767, 139918663784

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(7) = 15 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)(oo))  (oooo(o))
                                  (o(o)(o))  ((o)(ooo))
                                  (o(o(o)))  ((oo)(oo))
                                             (o(o)(oo))
                                             (o(o(oo)))
                                             (o(oo(o)))
                                             (oo(o)(o))
                                             (oo(o(o)))
                                             ((o)(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.

The same trees counted by leaves are A050381.
The locally disjoint version is A331872.
Matula-Goebel numbers of these trees are A331935.
Lone-child-avoiding rooted trees are A001678.
Cf. A000081, A000669, A198518, A289501, A291636, A306200, A320268, A330465, A330951, A331873, A331874, A331933, A331966.

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

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

Product_{k > 0} 1/(1 - x^k)^a(k) = A(x) + A(x)/x - x where A(x) = Sum_{k > 0} x^k a(k).

Euler transform is b(1) = 1, b(n > 1) = a(n) + a(n + 1).

Terms a(25) and beyond from Andrew Howroyd, Feb 09 2020

A001679 Number of series-reduced rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 0, 2, 2, 4, 6, 12, 20, 39, 71, 137, 261, 511, 995, 1974, 3915, 7841, 15749, 31835, 64540, 131453, 268498, 550324, 1130899, 2330381, 4813031, 9963288, 20665781, 42947715, 89410092, 186447559, 389397778, 814447067, 1705775653, 3577169927

Offset: 0

N. J. A. Sloane

nonn

Also known as homeomorphically irreducible rooted trees, or rooted trees without nodes of degree 2.

A rooted tree is lone-child-avoiding if no vertex has exactly one child, and topologically series-reduced if no vertex has degree 2. This sequence counts unlabeled topologically series-reduced rooted trees with n vertices. Lone-child-avoiding rooted trees with n - 1 vertices are counted by A001678. - Gus Wiseman, Jan 21 2020

			G.f. = 1 + x + x^2 + 2*x^4 + 2*x^5 + 4*x^6 + 6*x^7 + 12*x^8 + 20*x^9 + ...
From _Gus Wiseman_, Jan 21 2020: (Start)
The a(1) = 1 through a(8) = 12 unlabeled topologically series-reduced rooted trees with n nodes (empty n = 3 column shown as dot) are:
  o  (o)  .  (ooo)   (oooo)   (ooooo)    (oooooo)    (ooooooo)
             ((oo))  ((ooo))  ((oooo))   ((ooooo))   ((oooooo))
                              (oo(oo))   (oo(ooo))   (oo(oooo))
                              ((o(oo)))  (ooo(oo))   (ooo(ooo))
                                         ((o(ooo)))  (oooo(oo))
                                         ((oo(oo)))  ((o(oooo)))
                                                     ((oo(ooo)))
                                                     ((ooo(oo)))
                                                     (o(oo)(oo))
                                                     (oo(o(oo)))
                                                     (((oo)(oo)))
                                                     ((o(o(oo))))
(End)

D. G. Cantor, personal communication.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 62, Eq. (3.3.9).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183. MR0891613 (89a:05009). See p. 155. - _N. J. A. Sloane_, Apr 18 2014
F. Harary, G. Prins, The number of homeomorphically irreducible trees and other species, Acta Math. 101 (1959) 141-162, W(x,y) equation (9a).
N. J. A. Sloane, Illustration of initial terms
Eric Weisstein's World of Mathematics, Series-Reduced 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

Apart from initial term, same as A059123.
Cf. A000055 (trees by nodes), A000014 (homeomorphically irreducible trees by nodes), A000669 (homeomorphically irreducible planted trees by leaves), A000081 (rooted trees by nodes).
Cf. A246403.
The labeled version is A060313, with unrooted case A005512.
Matula-Goebel numbers of these trees are given by A331489.
Lone-child-avoiding rooted trees are counted by A001678(n + 1).
Cf. A004111, A060356, A198518, A254382, A291636, A330951, A331488, A331578.

with(powseries): with(combstruct): n := 30: Order := n+3: sys := {B = Prod(C,Z), S = Set(B,1 <= card), C = Union(Z,S)}:
G001678 := (convert(gfseries(sys,unlabeled,x)[S(x)], polynom)) * x^2: G0temp := G001678 + x^2:
G001679 := G0temp / x + G0temp - (G0temp^2+eval(G0temp,x=x^2))/(2*x): A001679 := 0,seq(coeff(G001679,x^i),i=1..n); # Ulrich Schimke (ulrschimke(AT)aol.com)
# adapted for Maple 16 or higher version by Vaclav Kotesovec, Jun 26 2014

terms = 37; (* F = G001678 *) F[] = 0; Do[F[x] = (x^2/(1 + x))*Exp[Sum[ F[x^k]/(k*x^k), {k, 1, j}]] + O[x]^j // Normal, {j, 1, terms + 1}];
G[x_] = 1 + ((1 + x)/x)*F[x] - (F[x]^2 + F[x^2])/(2*x) + O[x]^terms;
CoefficientList[G[x], x] (* Jean-François Alcover, Jan 12 2018 *)
urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]],{ptn,IntegerPartitions[n-1]}];
Table[Length[Select[urt[n],Length[#]!=2&&FreeQ[Z@@#,{}]&]],{n,15}] (* _Gus Wiseman, Jan 21 2020 *)

{a(n) = local(A); if( n<3, n>0, A = x / (1 - x^2) + x * O(x^n); for(k=3, n-1, A /= (1 - x^k + x * O(x^n))^polcoeff(A, k)); polcoeff( (1 + x)*A - x*(A^2 + subst(A, x, x^2)) / 2, n))};

G.f. = 1 + ((1+x)*f(x) - (f(x)^2+f(x^2))/2)/x where f(x) is g.f. for A001678 (homeomorphically irreducible planted trees by nodes).
a(n) ~ c * d^n / n^(3/2), where d = A246403 = 2.18946198566085056388702757711... and c = 0.4213018528699249210965028... . - Vaclav Kotesovec, Jun 26 2014
For n > 1, this sequence counts lone-child-avoiding rooted trees with n nodes and more than two branches, plus lone-child-avoiding rooted trees with n - 1 nodes. So for n > 1, a(n) = A331488(n) + A001678(n). - Gus Wiseman, Jan 21 2020

Additional comments from Michael Somos, Oct 10 2003

A331935 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 14, 16, 18, 21, 24, 26, 27, 28, 32, 36, 38, 39, 42, 46, 48, 49, 52, 54, 56, 57, 63, 64, 69, 72, 74, 76, 78, 81, 84, 86, 91, 92, 96, 98, 104, 106, 108, 111, 112, 114, 117, 122, 126, 128, 129, 133, 138, 144, 146, 147, 148, 152, 156, 159

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 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, two, and all nonprime numbers whose prime indices already belong to the sequence, where a prime index of n is a number m such that prime(m) divides n.

			The sequence of all semi-lone-child-avoiding rooted trees 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))
  21: ((o)(oo))
  24: (ooo(o))
  26: (o(o(o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  39: ((o)(o(o)))
  42: (o(o)(oo))
The sequence of terms together with their prime indices begins:
    1: {}              46: {1,9}             98: {1,4,4}
    2: {1}             48: {1,1,1,1,2}      104: {1,1,1,6}
    4: {1,1}           49: {4,4}            106: {1,16}
    6: {1,2}           52: {1,1,6}          108: {1,1,2,2,2}
    8: {1,1,1}         54: {1,2,2,2}        111: {2,12}
    9: {2,2}           56: {1,1,1,4}        112: {1,1,1,1,4}
   12: {1,1,2}         57: {2,8}            114: {1,2,8}
   14: {1,4}           63: {2,2,4}          117: {2,2,6}
   16: {1,1,1,1}       64: {1,1,1,1,1,1}    122: {1,18}
   18: {1,2,2}         69: {2,9}            126: {1,2,2,4}
   21: {2,4}           72: {1,1,1,2,2}      128: {1,1,1,1,1,1,1}
   24: {1,1,1,2}       74: {1,12}           129: {2,14}
   26: {1,6}           76: {1,1,8}          133: {4,8}
   27: {2,2,2}         78: {1,2,6}          138: {1,2,9}
   28: {1,1,4}         81: {2,2,2,2}        144: {1,1,1,1,2,2}
   32: {1,1,1,1,1}     84: {1,1,2,4}        146: {1,21}
   36: {1,1,2,2}       86: {1,14}           147: {2,4,4}
   38: {1,8}           91: {4,6}            148: {1,1,12}
   39: {2,6}           92: {1,1,9}          152: {1,1,1,8}
   42: {1,2,4}         96: {1,1,1,1,1,2}    156: {1,1,2,6}

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.

The enumeration of these trees by leaves is A050381.
The locally disjoint version A331873.
The enumeration of these trees by nodes is A331934.
The case with at most one distinct non-leaf branch of any vertex is A331936.
Lone-child-avoiding rooted trees are counted by A001678.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Cf. A000081, A000669, A007097, A061775, A196050, A320269, A330951, A331871, A331872, A331874, A331937, A331965.

mseQ[n_]:=n==1||n==2||!PrimeQ[n]&&And@@mseQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],mseQ]

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

A330943 Matula-Goebel numbers of singleton-reduced rooted trees.

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 12, 13, 14, 16, 18, 19, 21, 24, 26, 28, 32, 34, 36, 37, 38, 39, 42, 43, 48, 49, 52, 53, 54, 56, 57, 61, 63, 64, 68, 72, 73, 74, 76, 78, 82, 84, 86, 89, 91, 96, 98, 101, 102, 104, 106, 107, 108, 111, 112, 114, 117, 119, 122, 126, 128, 129, 131

Offset: 1

Gus Wiseman, Jan 13 2020

nonn

These trees are counted by A330951.
A rooted tree is singleton-reduced if no non-leaf node has all singleton branches, where a rooted tree is a singleton if its root has degree 1.
The Matula-Goebel number of a rooted tree is the product of primes of the Matula-Goebel numbers of its branches. This gives a bijective correspondence between positive integers and unlabeled rooted trees.
A prime index of n is a number m such that prime(m) divides n. A number belongs to this sequence iff it is 1 or its prime indices all belong to this sequence but are not all prime.

			The sequence of all singleton-reduced rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   7: ((oo))
   8: (ooo)
  12: (oo(o))
  13: ((o(o)))
  14: (o(oo))
  16: (oooo)
  18: (o(o)(o))
  19: ((ooo))
  21: ((o)(oo))
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  32: (ooooo)
  34: (o((oo)))
  36: (oo(o)(o))
  37: ((oo(o)))

The series-reduced case is A291636.
Unlabeled rooted trees are counted by A000081.
Numbers whose prime indices are not all prime are A330945.
Singleton-reduced rooted trees are counted by A330951.
Singleton-reduced phylogenetic trees are A000311.
The set S of numbers whose prime indices do not all belong to S is A324694.
Cf. A000669, A001678, A006450, A007097, A007821, A061775, A196050, A257994, A276625, A277098, A320628, A330944, A330948.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mgsingQ[n_]:=n==1||And@@mgsingQ/@primeMS[n]&&!And@@PrimeQ/@primeMS[n];
Select[Range[100],mgsingQ]

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

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A001678 Number of series-reduced planted trees with n nodes.

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

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A331934 Number of semi-lone-child-avoiding rooted trees with n unlabeled vertices.

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A001679 Number of series-reduced rooted trees with n nodes.

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A331935 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees.

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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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