A005629 Number of achiral trees with n nodes.
1, 1, 1, 2, 3, 5, 7, 14, 21, 40, 61, 118, 186, 355, 567, 1081, 1755, 3325, 5454, 10306, 17070, 32136, 53628, 100704, 169175, 316874, 535267, 1000524, 1698322, 3168500, 5400908, 10059823, 17211368, 32010736, 54947147, 102059572, 175702378
Offset: 1
Keywords
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- L. Bytautats and D. J. Klein, Alkane Isomer Combinatorics: Stereostructure enumeration and graph-invariant and molecular-property distributions, J. Chem. Inf. Comput. Sci 39 (1999) 803, Table 1.
- R. W. Robinson, F. Harary and A. T. Balaban, The numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (1976), 355-361.
- R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (1976), 355-361. (Annotated scanned copy)
- Index entries for sequences related to trees
Programs
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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 1 to 55 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: p[0]:=1: for n from 0 to 50 do > p[n+1]:=sum(s[k]*p[n-2*k],k=0..floor(n/2)) od:seq(p[j],j=0..45): P:=proc(n) if floor(n)=n then p[n] else 0 fi end:S:=proc(n) if floor(n)=n then s[n] else 0 fi end:t:=n->(P(n)+S(n/2)+S((n-1)/4))/2: seq(t(n),n=1..40); # here s[n]=A000625(n), p[n]=A005627(n). - Emeric Deutsch, Nov 21 2004
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Mathematica
nmax = 37; s[0] = s[1] = 1; s[_] = 0; 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, {n, 1, nmax}]; p[0] = 1; Do[p[n + 1] = Sum[s[k]*p[n - 2 k], {k, 0, Floor[n/2]}]; a[n + 1] = (p[n + 1] + s[(n + 1)/2] + s[n/4])/2, {n, 0, nmax}]; a[n_] := s[n] - p[n]; Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Jul 07 2024, after Maple code *)
Formula
a(n+1) = (p(n+1)+s((n+1)/2)+s(n/4))/2, where p(n)=A005627(n) and s(n)=A000625(n) (eq. (23) in the Robinson et al. reference). - Emeric Deutsch, Nov 21 2004
Extensions
Corrected and extended by Emeric Deutsch, Nov 21 2004