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A000625
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Number of n-node steric rooted ternary trees; number of n carbon alkyl radicals C(n)H(2n+1) taking stereoisomers into account
(Formerly M1402 N0546)
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17
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1, 1, 1, 2, 5, 11, 28, 74, 199, 551, 1553, 4436, 12832, 37496, 110500, 328092, 980491, 2946889, 8901891, 27012286, 82300275, 251670563, 772160922, 2376294040, 7333282754, 22688455980, 70361242924, 218679264772, 681018679604, 2124842137550, 6641338630714, 20792003301836
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| 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.
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REFERENCES
| 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.
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).
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LINKS
| 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
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FORMULA
| 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
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MAPLE
| 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);
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MATHEMATICA
| m = 31; c[0] = 1; gf[x_] = Sum[c[k] x^k, {k, 0, m}]; cc = Array[c, m]; coes = CoefficientList[ Series[gf[x] - 1 - (x*(gf[x]^3 + 2*gf[x^3])/3), {x, 0, m}], x] // Rest;
Prepend[cc /. Solve[ Thread[ coes == 0], cc][[1]], 1]
(* From Jean-François Alcover, Jun 24 2011 *)
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CROSSREFS
| Cf. A000598, A000602, A000620-A000624, A000628, A010732, A010733, A086194, A086200.
Sequence in context: A174145 A124016 A121398 * A202476 A127331 A040998
Adjacent sequences: A000622 A000623 A000624 * A000626 A000627 A000628
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Additional comments from Bruce Corrigan, Nov 04, 2002
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