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A005628 Number of chiral planted trees with n nodes.

%I A005628 M1641 #28 Jul 08 2025 16:32:44
%S A005628 0,0,0,0,2,6,20,60,176,510,1484,4314,12624,37126,109864,326958,978528,
%T A005628 2943384,8895792,27001378,82281216,251636434,772101086,2376186784,
%U A005628 7333094178,22688117658,70360646672,218678194238,681016789056
%N A005628 Number of chiral planted trees with n nodes.
%D A005628 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005628 R. W. Robinson, F. Harary and A. T. Balaban, <a href="http://dx.doi.org/10.1016/0040-4020(76)80049-X">The numbers of chiral and achiral alkanes and monosubstituted alkanes</a>, Tetrahedron 32 (1976), 355-361.
%H A005628 R. W. Robinson, F. Harary and A. T. Balaban, <a href="/A000625/a000625.pdf">Numbers of chiral and achiral alkanes and monosubstituted alkanes</a>, Tetrahedron 32 (3) (1976), 355-361. (Annotated scanned copy)
%H A005628 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H A005628 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A005628 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
%p A005628 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)
%t A005628 nmax = 28;
%t A005628 s[0] = s[1] = 1; s[_] = 0;
%t A005628 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}];
%t A005628 p[0] = 1;
%t A005628 Do[p[n+1] = Sum[s[k]*p[n-2*k], {k, 0, Floor[n/2]}], {n, 0, nmax}];
%t A005628 a[n_] := s[n] - p[n];
%t A005628 Table[a[n], {n, 0, nmax}] (* _Jean-François Alcover_, Jul 07 2024, after Maple code *)
%Y A005628 Cf. A000625, A005627.
%K A005628 nonn
%O A005628 0,5
%A A005628 _N. J. A. Sloane_
%E A005628 More terms from _Emeric Deutsch_, May 16 2004