A005627 -id:A005627 - cp's OEIS frontend

A005628 Number of chiral planted trees with n nodes.

0, 0, 0, 0, 2, 6, 20, 60, 176, 510, 1484, 4314, 12624, 37126, 109864, 326958, 978528, 2943384, 8895792, 27001378, 82281216, 251636434, 772101086, 2376186784, 7333094178, 22688117658, 70360646672, 218678194238, 681016789056

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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 rooted trees
Index entries for sequences related to trees

Cf. A000625, A005627.

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(s[n]-p[n],n=0..37); # here s[n]=A000625 and p[n]=A005627(n)

nmax = 28;
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]}], {n, 0, nmax}];
a[n_] := s[n] - p[n];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Jul 07 2024, after Maple code *)

a(n) = A000625(n)-A005627(n) (given as g(n)=s(n)-p(n) on p. 357 of the Robinson et al. paper). - Emeric Deutsch, May 16 2004

More terms from Emeric Deutsch, May 16 2004

A005629 Number of achiral trees with n nodes.

Original entry on oeis.org

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

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

Cf. A000625, A005627.

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

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 *)

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

Corrected and extended by Emeric Deutsch, Nov 21 2004

cp's OEIS Frontend

A005628 Number of chiral planted trees with n nodes.

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