0,4

Nodes are unlabeled, each node has out-degree <= 3.

Steric, or including stereoisomers, means that the children of a node are taken in a certain cyclic order. If the children are rotated it is still the same tree, but any other permutation yields a different tree. See A000598 for the analogous sequence with stereoisomers not counted.

Other descriptions of this sequence: steric planted trees with n nodes; total number of monosubstituted alkanes C(n)H(2n+1)-X with n carbon atoms.

Let the entries in the nine columns of Blair and Henze's Table I (JACS 54 (1932), p. 1098) be denoted by Ps(n), Pn(n), Ss(n), Sn(n), Ts(n), Tn(n), As(n), An(n), T(n) respectively (here P = Primary, S = Secondary, T = Tertiary, s = stereoisomers, n = non-stereoisomers and the last column T(n) gives total).

Then Ps (and As) = A000620, Pn (and An, Sn) = A000621, Ss = A000622, Ts = A000623, Tn = A000624, T = this sequence. Recurrences generating these sequences are given in the Maple program in A000620.

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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (1932), 1098-1105.

G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443, Eq. (25).

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; see p. 44.

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

T. D. Noe, Table of n, a(n) for n=0..200

Index entries for sequences related to rooted trees

Index entries for sequences related to trees

G.f. A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 + 28*x^6 + ... satisfies A(x) = 1 + x*(A(x)^3 + 2*A(x^3))/3.

a(0)=a(1)=1; a(n+1):=[2na(n/3)/3+sum(ja(j)sum(a(i)*a(n-j-i), i=0..n-j), j=1..n)]/n, (n>=2), where a(k)=0 if k not an integer (essentially eq (4) in the Robinson et al. paper). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 16 2004

A := 1; f := proc(n) global A; coeff(series( 1+(1/3)*x*(A^3+2*subs(x=x^3, A)), x, n+1), x, n); end; for n from 1 to 50 do A := series(A+f(n)*x^n, x, n +1); od: A;

a[0]:=1: a[1]:=1: for n from 0 to 50 do a[n+1/3]:=0 od:for n from 0 to 50 do a[n+2/3]:=0 od:for n from 1 to 35 do a[n+1]:=(2*n/3*a[n/3]+sum(j*a[j]*sum(a[i]*a[n-j-i], i=0..n-j), j=1..n))/n od:seq(a[j], j=0..31);

Cf. A000598, A000602, A000620-A000624, A000628, A010732, A010733, A086194, A086200.

Sequence in context: A055227 A124016 A121398 this_sequence A127331 A040998 A117719

Adjacent sequences: A000622 A000623 A000624 this_sequence A000626 A000627 A000628

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Bruce Corrigan, Nov 04, 2002