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%I M0732 N0274 #65 Dec 06 2020 08:31:16
%S 1,1,1,1,2,3,5,11,24,55,136,345,900,2412,6563,18127,50699,143255,
%T 408429,1173770,3396844,9892302,28972080,85289390,252260276,749329719,
%U 2234695030,6688893605,20089296554,60526543480,182896187256,554188210352,1683557607211,5126819371356,15647855317080,47862049187447,146691564302648,450451875783866,1385724615285949
%N Number of n-node unrooted steric quartic trees; number of n-carbon alkanes C(n)H(2n+2) taking stereoisomers into account.
%C Trees are unrooted; nodes are unlabeled and have degree <= 4.
%C 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.
%C Has also been described as steric planted trees (paraffins) with n nodes.
%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 290.
%D 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. 278-281.
%D J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H C. M. Blair and H. R. Henze, <a href="http://dx.doi.org/10.1021/ja01343a044">The number of stereoisomeric and non-stereoisomeric paraffin hydrocarbons</a>, J. Amer. Chem. Soc., 54 (1932), 1538-1545.
%H C. M. Blair and H. R. Henze, <a href="/A000626/a000626.pdf">The number of stereoisomeric and non-stereoisomeric paraffin hydrocarbons</a>, J. Amer. Chem. Soc., 54 (4) (1932), 1538-1545. (Annotated scanned copy)
%H L. Bytautats and D. J. Klein, <a href="https://doi.org/10.1021/ci990021g">Alkane Isomere Combinatorics: Stereostructure enumeration and graph-invariant and molecylar-property distributions</a>, J. Chem. Inf. Comput. Sci 39 (1999) 803-818, Table 1.
%H P. Leroux and B. Miloudi, <a href="http://www.labmath.uqam.ca/~annales/volumes/16-1/PDF/053-080.pdf">Généralisations de la formule d'Otter</a>, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992.
%H P. Leroux and B. Miloudi, <a href="/A000081/a000081_2.pdf">Généralisations de la formule d'Otter</a>, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
%H R. C. Read, <a href="/A000598/a000598.pdf">The Enumeration of Acyclic Chemical Compounds</a>, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [Annotated scanned copy] See p. 44.
%H R. W. Robinson, F. Harary and A. T. Balaban, <a href="http://dx.doi.org/10.1016/0040-4020(76)80049-X">The numbers of chiral and achiral alkanes and monosubstituted alkanes</a>, Tetrahedron 32 (1976), 355-361.
%H R. W. Robinson, F. Harary and A. T. Balaban, <a href="/A000625/a000625.pdf">Numbers of chiral and achiral alkanes and monosubstituted alkanes</a>, Tetrahedron 32 (3) (1976), 355-361. (Annotated scanned copy)
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F Blair and Henze give recurrence (see the Maple code).
%F For even n a(n) = A086194(n) + A086200(n/2), for odd n a(n) = A086194(n).
%p 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[n-j-k],k=0..n-j),j=1..n))/n od:for n from 0 to 50 do q[n]:=sum(s[i]*s[n-i],i=0..n) od:for n from 0 to 50 do q[n-1/2]:=0 od:for n from 0 to 40 do f:=n->(3*s[n]+2*s[n/2]+q[(n-1)/2]-q[n]+2*sum(s[j]*s[n-3*j-1],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_
%t 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[n-j-k], {k, 0, n-j}], {j, 1, n}])/n]; For[n=0, n <= max, n++, q[n] = Sum[s[i]*s[n-i], {i, 0, n}]]; For[n=0, n <= max, n++, q[n-1/2]=0]; f[n_] := (3*s[n] + 2*s[n/2] + q[(n-1)/2] - q[n] + 2*Sum[s[j]*s[n-3*j-1], {j, 0, n/3}])/4; Table[f[n], {n, 0, max}] (* _Jean-François Alcover_, Dec 29 2014, after _Emeric Deutsch_ *)
%Y Equals A000626 + A000627.
%Y Cf. A000598, A000602, A000625, A010372, A010373, A086194, A086200.
%K nonn,easy,nice
%O 0,5
%A _N. J. A. Sloane_
%E Additional comments from Steve Strand (snstrand(AT)comcast.net), Aug 20 2003
%E More terms from _Emeric Deutsch_, May 16 2004
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