A005627 Number of achiral planted trees with n nodes. (Formerly M0698)

1, 1, 1, 2, 3, 5, 8, 14, 23, 41, 69, 122, 208, 370, 636, 1134, 1963, 3505, 6099, 10908, 19059, 34129, 59836, 107256, 188576, 338322, 596252, 1070534, 1890548, 3396570, 6008908, 10801816, 19139155, 34422537, 61074583, 109894294, 195217253

Offset

0,4

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=0..36.

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.

Formula

a(0)=1, a(n+1):=sum(s(k)*a(n-2*k), k=0..floor(n/2)) (n>=0), where s(n)=A000625(n) (this is eq. (15) in the Robinson et al. paper). - Emeric Deutsch, May 16 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:a[0]:=1: for n from 0 to 50 do a[n+1]:=sum(s[k]*a[n-2*k], k=0..floor(n/2)) od:seq(a[j], j=0..45); # here s[n]=A000625(n).

Mathematica

nmax = 36;
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}];
a[0] = a[1] = 1;
Do[a[n+1] = Sum[s[k]*a[n-2*k], {k, 0, Floor[n/2]}], {n, 1, nmax}];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Jul 07 2024, after Maple code *)

Crossrefs

Cf. A000625.

Sequence in context: A039828 A357303 A246360 * A191794 A191388 A194850

Adjacent sequences: A005624 A005625 A005626 * A005628 A005629 A005630

Keyword

nonn,changed

Author

N. J. A. Sloane

Extensions

More terms from Emeric Deutsch, May 16 2004

Status

approved