

A000628


Number of nnode unrooted steric quartic trees; number of ncarbon alkanes C(n)H(2n+2) taking stereoisomers into account.
(Formerly M0732 N0274)


13



1, 1, 1, 1, 2, 3, 5, 11, 24, 55, 136, 345, 900, 2412, 6563, 18127, 50699, 143255, 408429, 1173770, 3396844, 9892302, 28972080, 85289390, 252260276, 749329719, 2234695030, 6688893605, 20089296554, 60526543480, 182896187256, 554188210352, 1683557607211, 5126819371356, 15647855317080, 47862049187447, 146691564302648, 450451875783866, 1385724615285949
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OFFSET

0,5


COMMENTS

Trees are unrooted; nodes are unlabeled and have degree <= 4.
Regarding stereoisomers as different means that only the alternating group A_4 acts at each node, not the full symmetric group S_4. See A000602 for the analogous sequence when stereoisomers are not counted as different.
Has also been described as steric planted trees (paraffins) with n nodes.


REFERENCES

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and TreeLike Structures, Camb. 1998, p. 290.
R. Davies and P. J. Freyd, C_{167}H_{336} is The Smallest Alkane with More Realizable Isomers than the Observable Universe has Particles, Journal of Chemical Education, Vol. 66, 1989, pp. 278281.
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, WileyVCH, 2005.
R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 2561 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976; see p. 44.
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).


LINKS

Table of n, a(n) for n=0..38.
C. M. Blair and H. R. Henze, The number of stereoisomeric and nonstereoisomeric paraffin hydrocarbons, J. Amer. Chem. Soc., 54 (1932), 15381545.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 5380, 1992.
R. W. Robinson, F. Harary and A. T. Balaban, The numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (1976), 355361.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees


FORMULA

Blair and Henze give recurrence (see the Maple code).
For even n a(n) = A086194(n) + A086200(n/2), for odd n a(n) = A086194(n).


MAPLE

s[0]:=1:s[1]:=1:for n from 0 to 60 do s[n+1/3]:=0 od:for n from 0 to 60 do s[n+2/3]:=0 od:for n from 0 to 60 do s[n+1/4]:=0 od:for n from 0 to 60 do s[n+1/2]:=0 od:for n from 0 to 60 do s[n+3/4]:=0 od:s[ 1]:=0:for n from 1 to 50 do s[n+1]:=(2*n/3*s[n/3]+sum(j*s[j]*sum(s[k]*s[njk], k=0..nj), j=1..n))/n od:for n from 0 to 50 do q[n]:=sum(s[i]*s[ni], i=0..n) od:for n from 0 to 50 do q[n1/2]:=0 od:for n from 0 to 40 do f:=n>(3*s[n]+2*s[n/2]+q[(n1)/2]q[n]+2*sum(s[j]*s[n3*j1], j=0..n/3))/4 od:seq(f(n), n=0..38); # the formulas for s[n+1] and f(n) are from eq.(4) and (12), respectively, of the Robinson et al. paper; s[n]=A000625(n), f(n)=A000628(n); q[n] is the convolution of s[n] with itself; # Emeric Deutsch


MATHEMATICA

max = 40; s[0] = s[1] = 1; s[_] = 0; For[n=1, n <= max, n++, s[n+1] = (2*n/3*s[n/3] + Sum[j*s[j]*Sum[s[k]*s[njk], {k, 0, nj}], {j, 1, n}])/n]; For[n=0, n <= max, n++, q[n] = Sum[s[i]*s[ni], {i, 0, n}]]; For[n=0, n <= max, n++, q[n1/2]=0]; f[n_] := (3*s[n] + 2*s[n/2] + q[(n1)/2]  q[n] + 2*Sum[s[j]*s[n3*j1], {j, 0, n/3}])/4; Table[f[n], {n, 0, max}] (* JeanFrançois Alcover, Dec 29 2014, after Emeric Deutsch *)


CROSSREFS

Equals A000626 + A000627.
Cf. A000598, A000602, A000625, A010372, A010373, A086194, A086200.
Sequence in context: A176499 A175234 A060696 * A006888 A009589 A098179
Adjacent sequences: A000625 A000626 A000627 * A000629 A000630 A000631


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

Additional comments from Steve Strand (snstrand(AT)comcast.net), Aug 20 2003
More terms from Emeric Deutsch, May 16 2004


STATUS

approved



