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A000598 Number of rooted ternary trees with n nodes; number of n-carbon alkyl radicals C(n)H(2n+1) ignoring stereoisomers.
(Formerly M1146 N0436 N1341)
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 (list; graph; refs; listen; history; text; internal format)



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


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.

K. Grützmann, S. Böcker, S. Schuster, Combinatorics of aliphatic amino acids, Naturwissenschaften, Vol. 98, No. 1, 79-86, 2011

Knop, Mueller, Szymanski and Trinajstich, Computer generation of certain classes of molecules.

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 ..., 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).

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. 379-39O, 1983.


N. J. A. Sloane and 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 (4 (1988), 304-313.

Frederic Chyzak, Enumerating alcohols and other classes of chemical molecules

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 478

H. R. Henze, C. M. Blair, The number of structurally isomeric alcohols of the methanol series, J. Amer. Chem. Soc., 53 (8) (1931), 3042-3046.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1

G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443; Table I, line 2.

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.

R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (1976), 355-361.

N. J. A. Sloane, Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678

Wikipedia, Polya's enumeration theorem

Index entries for sequences related to rooted trees

Index entries for sequences related to trees


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


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; For[gf = 0; i = 0, i <= m, i++, gf = Series[1 + z*(gf^3/6 + (gf /. z -> z^2)*gf/2 + (gf /. z -> z^3)/3), {z, 0, m + 1}] // Normal]; CoefficientList[gf, z][[1 ;; m]] (* Jean-François Alcover, Sep 23 2014, updated Apr 18 2016 *)


Cf. A000599, A000600, A000602, A000625, A000628, A000678, A010372, A010373, A086194, A086200, A261340.

Sequence in context: A241671 A036375 A036376 * A003008 A185352 A054199

Adjacent sequences:  A000595 A000596 A000597 * A000599 A000600 A000601




N. J. A. Sloane.


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



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Last modified February 21 11:31 EST 2017. Contains 282442 sequences.