A005627 Number of achiral planted trees with n nodes.
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
Keywords
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- 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.
Crossrefs
Cf. A000625.
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: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).
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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 *)
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
Extensions
More terms from Emeric Deutsch, May 16 2004