A298204 -id:A298204 - cp's OEIS frontend

A000598 Number of rooted ternary trees with n nodes; number of n-carbon alkyl radicals C(n)H(2n+1) ignoring stereoisomers.

1, 1, 1, 2, 4, 8, 17, 39, 89, 211, 507, 1238, 3057, 7639, 19241, 48865, 124906, 321198, 830219, 2156010, 5622109, 14715813, 38649152, 101821927, 269010485, 712566567, 1891993344, 5034704828, 13425117806, 35866550869, 95991365288, 257332864506, 690928354105

Offset: 0

N. J. A. Sloane

Number of unlabeled rooted trees in which each node has out-degree <= 3.
Ignoring stereoisomers means that the children of a node are unordered. They can be permuted in any way and it is still the same tree. See A000625 for the analogous sequence with stereoisomers counted.
In alkanes every carbon has valence exactly 4 and every hydrogen has valence exactly 1. But the trees considered here are just the carbon "skeletons" (with all the hydrogen atoms stripped off) so now each carbon bonds to 1 to 4 other carbons. The out-degree is then <= 3.
Other descriptions of this sequence: quartic planted trees with n nodes; ternary rooted trees with n nodes and height at most 3.
The number of aliphatic amino acids with n carbon atoms in the side chain, and no rings or double bonds, has the same growth as this sequence. - Konrad Gruetzmann, Aug 13 2012

			From _Joerg Arndt_, Feb 25 2017: (Start)
The a(5) = 8 rooted trees with 5 nodes and out-degrees <= 3 are:
:         level sequence    out-degrees (dots for zeros)
:     1:  [ 0 1 2 3 4 ]    [ 1 1 1 1 . ]
:  O--o--o--o--o
:
:     2:  [ 0 1 2 3 3 ]    [ 1 1 2 . . ]
:  O--o--o--o
:        .--o
:
:     3:  [ 0 1 2 3 2 ]    [ 1 2 1 . . ]
:  O--o--o--o
:     .--o
:
:     4:  [ 0 1 2 3 1 ]    [ 2 1 1 . . ]
:  O--o--o--o
:  .--o
:
:     5:  [ 0 1 2 2 2 ]    [ 1 3 . . . ]
:  O--o--o
:     .--o
:     .--o
:
:     6:  [ 0 1 2 2 1 ]    [ 2 2 . . . ]
:  O--o--o
:     .--o
:  .--o
:
:     7:  [ 0 1 2 1 2 ]    [ 2 1 . 1 . ]
:  O--o--o
:  .--o--o
:
:     8:  [ 0 1 2 1 1 ]    [ 3 1 . . . ]
:  O--o--o
:  .--o
:  .--o
(End)

N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 62 (quoting Cayley, who is wrong).
A. Cayley, On the mathematical theory of isomers, Phil. Mag. vol. 67 (1874), 444-447 (a(6) is wrong).
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
R. A. Fisher, Contributions to Mathematical Statistics, Wiley, 1950, 41.397.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 529.
Handbook of Combinatorics, North-Holland '95, p. 1963.
Knop, Mueller, Szymanski and Trinajstich, Computer generation of certain classes of molecules.
D. Perry, The number of structural isomers ..., J. Amer. Chem. Soc. 54 (1932), 2918-2920.
G. Polya, Mathematical and Plausible Reasoning, Vol. 1 Prob. 4 pp. 85; 233.
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).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (first 200 terms from N. J. A. Sloane)
Jean-François Alcover, Mathematica program translated from N. J. A. Sloane's Maple program for A000022, A000200, A000598, A000602, A000678.
A. T. Balaban, J. W. Kennedy and L. V. Quintas, The number of alkanes having n carbons and a longest chain of length d, J. Chem. Education, 65 (1988), 304-313.
A. Cayley, On the mathematical theory of isomers, Phil. Mag. vol. 67 (1874), 444-447 (a(6) is wrong). (Annotated scanned copy)
Frederic Chyzak, Enumerating alcohols and other classes of chemical molecules.
Maximilian Fichtner, K. Voigt, and S. Schuster, The tip and hidden part of the iceberg: Proteinogenic and non-proteinogenic aliphatic amino acids, Biochimica et Biophysica Acta (BBA)-General, 2016, Volume 1861, Issue 1, Part A, January 2017, pp. 3258-3269.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 478.
K. Grützmann, S. Böcker, and S. Schuster, Combinatorics of aliphatic amino acids, Naturwissenschaften, Vol. 98, No. 1, 79-86, 2011.
H. R. Henze and C. M. Blair, The number of structurally isomeric alcohols of the methanol series, J. Amer. Chem. Soc., 53 (8) (1931), 3042-3046.
H. R. Henze and C. M. Blair, The number of structurally isomeric alcohols of the methanol series, J. Amer. Chem. Soc., 53 (8) (1931), 3042-3045. (Annotated scanned copy)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).
D. Perry, The number of structural isomers of certain homologs of methane and methanol, J. Amer. Chem. Soc. 54 (1932), 2918-2920. [Gives a(60) correctly.] (Annotated scanned copy)
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; Table I, line 2.
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; Table I, line 2. (Annotated scanned copy)
R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [Annotated scanned copy] See p. 20, Eq. (G); p. 27, Eq. 2.1.
Marko Riedel, Chris Grossack, Counting trees with a degree constraint, Math StackExchange.
R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (1976), 355-361.
R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (1976), 355-361. (Annotated scanned copy)
Hugo Schiff, Zur Statistik chemischer Verbindungen, Berichte der Deutschen Chemischen Gesellschaft, Vol. 8, pp. 1542-1547, 1875. [Annotated scanned copy]
N. J. A. Sloane, Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678.
N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer Generation of Isomeric Structures, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.
Wikipedia, Polya's enumeration theorem.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000599, A000600, A000602, A000625, A000628, A000678, A010372, A010373, A086194, A086200, A261340.
Cf. A292553, A292554, A292555, A292556.
Cf. A000081, A001190, A014591, A032305, A295461, A298118, A298120, A298204, A298422, A298426.
Column k=3 of A299038.

N := 45; G000598 := 0: i := 0: while i<(N+1) do G000598 := series(1+z*(G000598^3/6+subs(z=z^2,G000598)*G000598/2+subs(z=z^3,G000598)/3)+O(z^(N+1)),z,N+1): t[ i ] := G000598: i := i+1: od: A000598 := n->coeff(G000598,z,n);
# Another Maple program for g.f. G000598:
G000598 := 1; f := proc(n) global G000598; coeff(series(1+(1/6)*x*(G000598^3+3*G000598*subs(x=x^2,G000598)+2*subs(x=x^3,G000598)),x, n+1),x,n); end; for n from 1 to 50 do G000598 := series(G000598+f(n)*x^n,x,n+1); od; G000598;
spec := [S, {Z=Atom, S=Union(Z, Prod(Z, Set(S, card=3)))}, unlabeled]: [seq(combstruct[count](spec, size=n), n=0..20)];

m = 45; Clear[f]; f[1, x_] := 1+x; f[n_, x_] := f[n, x] = Expand[1+x*(f[n-1, x]^3/6 + f[n-1, x^2]*f[n-1, x]/2 + f[n-1, x^3]/3)][[1 ;; n]]; Do[f[n, x], {n, 2, m}]; CoefficientList[f[m, x], x]
(* second program (after N. J. A. Sloane): *)
m = 45; gf[] = 0; Do[gf[z] = 1 + z*(gf[z]^3/6 + gf[z^2]*gf[z]/2 + gf[z^3]/3) + O[z]^m // Normal, m]; CoefficientList[gf[z], z]  (* Jean-François Alcover, Sep 23 2014, updated Jan 11 2018 *)
b[0, i_, t_, k_] = 1; m = 3; (* m = maximum children *)
b[n_,i_,t_,k_]:= b[n,i,t,k]= If[i<1,0,
  Sum[Binomial[b[i-1, i-1, k, k] + j-1, j]*
  b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]];
Join[{1},Table[b[n-1, n-1, m, m], {n, 1, 35}]] (* Robert A. Russell, Dec 27 2022 *)

seq(n)={my(g=O(x)); for(n=1, n, g = 1 + x*(g^3/6 + subst(g,x,x^2)*g/2 + subst(g,x,x^3)/3) + O(x^n)); Vec(g)} \\ Andrew Howroyd, May 22 2018

def seq(n):
    B = PolynomialRing(QQ, 't', n+1);t = B.gens()
    R. = B[[]]
    T = sum([t[i] * z^i for i in range(1,n+1)]) + O(z^(n+1))
    lhs, rhs = T, 1 + z/6 * (T(z)^3 + 3*T(z)*T(z^2) + 2*T(z^3))
    I = B.ideal([lhs.coefficients()[i] - rhs.coefficients()[i] for i in range(n)])
    return [I.reduce(t[i]) for i in range(1,n+1)]
seq(33) # Chris Grossack, Mar 31 2025

G.f. A(x) satisfies A(x) = 1 + (1/6)*x*(A(x)^3 + 3*A(x)*A(x^2) + 2*A(x^3)).

a(n) ~ c * d^n / n^(3/2), where d = 1/A261340 = 2.8154600331761507465266167782426995425365065396907..., c = 0.517875906458893536993162356992854345458168348098... . - Vaclav Kotesovec, Aug 15 2015

Additional comments from Steve Strand (snstrand(AT)comcast.net), Aug 20 2003

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

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}]

A303023 Number of anti-binary (no binary branchings) unlabeled rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 32, 66, 139, 297, 642, 1404, 3097, 6888, 15428, 34770, 78785, 179397, 410264, 941935, 2170275, 5016604, 11630024, 27034824, 63000261, 147148341, 344419767, 807746487, 1897829065, 4466643367, 10529301944, 24858143953, 58769113863

Offset: 1

Gus Wiseman, Aug 15 2018

nonn

			The a(6) = 8 rooted trees:
  (((((o)))))
  (((ooo)))
  ((oo(o)))
  (oo((o)))
  (o(o)(o))
  ((oooo))
  (ooo(o))
  (ooooo)

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A000081, A000598, A001190, A001678, A102403, A292050, A298204, A298422.

Cf. A303022, A303024, A303025, A303026, A303027, A318520.

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=1, 0, 1), `if`(i<1, 0,
      add(b(n-i*j, i-1, max(0, t-j))*binomial(a(i)+j-1, j), j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, b(n-1$2, 3)):
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 27 2018

burt[n_]:=burt[n]=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[burt/@c]],{c,Select[IntegerPartitions[n-1],Length[#]!=2&]}]];
Table[Length[burt[n]],{n,20}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 1, 0, 1], If[i < 1, 0, Sum[b[n-i*j, i-1, Max[0, t-j]]*Binomial[a[i]+j-1, j], {j, 0, n/i}]]];
a[n_] := If[n < 2, n, b[n-1, n-1, 3]];
Array[a, 50] (* Jean-François Alcover, May 16 2021, after Alois P. Heinz *)

a(24)-a(34) from Alois P. Heinz, Aug 27 2018

A303025 Number of series-reduced anti-binary (no unary or binary branchings) unlabeled rooted trees with n nodes.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 3, 4, 7, 11, 17, 28, 46, 74, 123, 205, 341, 571, 964, 1629, 2764, 4707, 8040, 13766, 23639, 40681, 70163, 121256, 209960, 364168, 632694, 1100906, 1918375, 3347346, 5848271, 10229977, 17915018, 31407088, 55116661, 96818589, 170229939

Offset: 1

Gus Wiseman, Aug 15 2018

nonn

			The a(10) = 7 rooted trees:
  (oo(oo(ooo)))
  (o(ooo)(ooo))
  (oo(oooooo))
  (ooo(ooooo))
  (oooo(oooo))
  (ooooo(ooo))
  (ooooooooo)

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A000081, A000598, A001190, A001678, A102403, A298204, A298422.

Cf. A303022, A303023, A303024, A303026, A303027.

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0), `if`(i<1, 0,
      add(b(n-i*j, i-1, max(0, t-j))*binomial(a(i)+j-1, j), j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, b(n-1$2, 3)):
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 27 2018

zurt[n_]:=zurt[n]=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[zurt/@c]],{c,Select[IntegerPartitions[n-1],Length[#]>2&]}]];
Table[Length[zurt[n]],{n,20}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0], If[i < 1, 0, Sum[b[n-i*j, i - 1, Max[0, t-j]]*Binomial[a[i]+j-1, j], {j, 0, n/i}]]];
a[n_] :=  If[n < 2, n, b[n-1, n-1, 3]];
Array[a, 50] (* Jean-François Alcover, May 17 2021, after Alois P. Heinz *)

a(36)-a(42) from Alois P. Heinz, Aug 27 2018

A320270 Number of unlabeled balanced semi-binary rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 7, 10, 13, 19, 25, 35, 46, 65, 88, 124, 171, 242, 334, 470, 653, 921, 1287, 1822, 2565, 3640, 5144, 7311, 10360, 14734, 20918, 29781, 42361, 60389, 86069, 122893, 175479, 250922, 358863

Offset: 1

Gus Wiseman, Oct 08 2018

nonn

An unlabeled rooted tree is semi-binary if all out-degrees are <= 2, and balanced if all leaves are the same distance from the root. The number of semi-binary trees with n nodes is equal to the number of binary trees with n+1 leaves; see A001190.

			The a(1) = 1 through a(7) = 6 balanced semi-binary rooted trees:
  o  (o)  (oo)   ((oo))   (((oo)))   ((o)(oo))    ((oo)(oo))
          ((o))  (((o)))  ((o)(o))   ((((oo))))   (((o)(oo)))
                          ((((o))))  (((o)(o)))   (((((oo)))))
                                     (((((o)))))  ((((o)(o))))
                                                  (((o))((o)))
                                                  ((((((o))))))

Gus Wiseman, The a(13) = 35 balanced semi-binary rooted trees.
Gus Wiseman, The a(15) = 65 balanced semi-binary rooted trees.
Gus Wiseman, The a(16) = 88 balanced semi-binary rooted trees.
Gus Wiseman, The a(18) = 171 balanced semi-binary rooted trees.

Cf. A001190, A048816, A079500, A111299, A292050, A298204, A301345, A320271.

saur[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[saur/@ptn]],SameQ@@Length/@Position[#,{}]&],{ptn,Select[IntegerPartitions[n-1],Length[#]<=2&]}]];
Table[Length[saur[n]],{n,20}]

A298207 Numbers that are a product of zero, one, or three (not necessarily distinct) prime numbers.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, 27, 28, 29, 30, 31, 37, 41, 42, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 66, 67, 68, 70, 71, 73, 75, 76, 78, 79, 83, 89, 92, 97, 98, 99, 101, 102, 103, 105, 107, 109, 110, 113, 114, 116, 117, 124, 125, 127, 130

Offset: 1

Gus Wiseman, Jan 14 2018

nonn

			1 is a product of zero primes so is in the sequence.
6 = 2 * 3 is a product of two primes so is not in the sequence.
12 = 2 * 2 * 3 is a product of three primes so is in the sequence.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358, A001399, A008578, A014591, A014612, A026424, A028260, A298120, A298204, A298205.

Subsequence of A037144. - Felix Fröhlich, Jan 15 2018

a:= proc(n) option remember; local k; for k
      from 1+`if`(n=1, 0, a(n-1)) while not
      numtheory[bigomega](k) in {0, 1, 3} do od; k
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Jan 15 2018

Select[Range[200],MemberQ[{0,1,3},PrimeOmega[#]]&]

is(n) = my(v=[0, 1, 3]); #setintersect([bigomega(n)], v)==1 \\ Felix Fröhlich, Jan 15 2018

Equals {A008578} union {A014612}.

Equals {A037144} minus {A001358}.

A298205 Matula-Goebel numbers of rooted trees in which all outdegrees are either 0, 1, or 3.

Original entry on oeis.org

1, 2, 3, 5, 8, 11, 12, 18, 19, 20, 27, 30, 31, 37, 44, 45, 50, 61, 66, 67, 71, 75, 76, 99, 103, 110, 113, 114, 124, 125, 127, 148, 157, 165, 171, 186, 190, 193, 197, 222, 229, 242, 244, 268, 275, 279, 283, 284, 285, 310, 317, 331, 333, 353, 363, 366, 370, 379

Offset: 1

Gus Wiseman, Jan 14 2018

nonn

			Sequence of rooted 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)))))
37 ((oo(o)))
44 (oo(((o))))
45 ((o)(o)((o)))
50 (o((o))((o)))

Cf. A000081, A000598, A014591, A026424, A032305, A061775, A111299, A276625, A292050, A295461, A297571, A298126, A298118, A298120, A298204, A298207.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stQ[n_]:=Or[n===1,With[{m=primeMS[n]},MemberQ[{1,3},Length[m]]&&And@@stQ/@m]];
Select[Range[10000],stQ]

A317097 Number of rooted trees with n nodes where the number of distinct branches under each node is <= 2.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 46, 106, 248, 583, 1393, 3343, 8111, 19801, 48719, 120489, 299787, 749258, 1881216, 4741340, 11993672, 30436507, 77471471, 197726053, 505917729, 1297471092, 3334630086, 8587369072, 22155278381, 57259037225, 148222036272, 384272253397

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

There can be more than two branches as long as there are not three distinct branches.

			The a(5) = 9 trees:
  ((((o))))
  (((oo)))
  ((o(o)))
  ((ooo))
  (o((o)))
  (o(oo))
  ((o)(o))
  (oo(o))
  (oooo)

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

Cf. A000081, A000598, A001190, A003238, A055277, A111299, A292050, A298204, A301344, A317098.

semisameQ[u_]:=Length[Union[u]]<=2;
nms[n_]:=nms[n]=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],semisameQ],{ptn,IntegerPartitions[n-1]}]];
Table[Length[nms[n]],{n,10}]

seq(n)={my(v=vector(n)); v[1]=1; for(n=1, n-1, v[n+1]=sum(k=1, n-1, sumdiv(k, d, v[d])*sumdiv(n-k, d, v[d])/2) + sumdiv(n, d, v[n/d]*(1 - (d-1)/2)) ); v} \\ Andrew Howroyd, Aug 28 2018

Terms a(20) and beyond from Andrew Howroyd, Aug 28 2018

A317098 Number of series-reduced rooted trees with n unlabeled leaves where the number of distinct branches under each node is <= 2.

Original entry on oeis.org

1, 1, 2, 5, 12, 31, 80, 214, 576, 1595, 4448, 12625, 36146, 104662, 305251, 897417, 2654072, 7895394, 23601441, 70871693, 213660535, 646484951, 1962507610, 5975425743, 18243789556, 55841543003, 171320324878, 526738779846, 1622739134873, 5008518981670

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

There can be more than two branches as long as there are not three distinct branches.

			The a(5) = 12 trees:
  (o(o(o(oo))))
  (o(o(ooo)))
  (o((oo)(oo)))
  (o(oo(oo)))
  (o(oooo))
  ((oo)(o(oo)))
  ((oo)(ooo))
  (oo(o(oo)))
  (oo(ooo))
  (o(oo)(oo))
  (ooo(oo))
  (ooooo)

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

Cf. A000081, A000598, A001190, A055277, A111299, A292050, A298204, A301344, A317097.

semisameQ[u_]:=Length[Union[u]]<=2;
nms[n_]:=nms[n]=If[n==1,{{1}},Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],semisameQ],{ptn,Rest[IntegerPartitions[n]]}]];
Table[Length[nms[n]],{n,10}]

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

Terms a(21) and beyond from Andrew Howroyd, Aug 19 2018

cp's OEIS Frontend

A000598 Number of rooted ternary trees with n nodes; number of n-carbon alkyl radicals C(n)H(2n+1) ignoring stereoisomers.

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

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

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A303023 Number of anti-binary (no binary branchings) unlabeled rooted trees with n nodes.

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A303025 Number of series-reduced anti-binary (no unary or binary branchings) unlabeled rooted trees with n nodes.

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A320270 Number of unlabeled balanced semi-binary rooted trees with n nodes.

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A298207 Numbers that are a product of zero, one, or three (not necessarily distinct) prime numbers.

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