A005629 Number of achiral trees with n nodes. (Formerly M0677)

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

1,4

References

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=1..37.

L. Bytautats and D. J. Klein, Alkane Isomere Combinatorics: Stereostructure enumeration and graph-invariant and molecylar-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

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

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

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 *)

Crossrefs

Cf. A000625, A005627.

Sequence in context: A114625 A046808 A097799 * A028304 A324840 A316475

Adjacent sequences: A005626 A005627 A005628 * A005630 A005631 A005632

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Corrected and extended by Emeric Deutsch, Nov 21 2004

Status

approved