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A000628
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Number of n-node unrooted steric quartic trees; number of n-carbon alkanes C(n)H(2n+2) taking stereoisomers into account.
(Formerly M0732 N0274)
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15
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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
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0,5
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COMMENTS
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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.
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REFERENCES
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F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like 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. 278-281.
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
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
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FORMULA
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Blair and Henze give recurrence (see the Maple code).
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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Additional comments from Steve Strand (snstrand(AT)comcast.net), Aug 20 2003
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STATUS
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approved
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