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A005627 Number of achiral planted trees with n nodes.

%I A005627 M0698 #42 Jul 07 2024 03:34:17
%S A005627 1,1,1,2,3,5,8,14,23,41,69,122,208,370,636,1134,1963,3505,6099,10908,
%T A005627 19059,34129,59836,107256,188576,338322,596252,1070534,1890548,
%U A005627 3396570,6008908,10801816,19139155,34422537,61074583,109894294,195217253
%N A005627 Number of achiral planted trees with n nodes.
%D A005627 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005627 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 A005627 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 A005627 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>.
%H A005627 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>.
%F A005627 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
%p 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: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).
%t A005627 nmax = 36;
%t A005627 s[0] = s[1] = 1; s[_] = 0;
%t A005627 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 A005627 a[0] = a[1] = 1;
%t A005627 Do[a[n+1] = Sum[s[k]*a[n-2*k], {k, 0, Floor[n/2]}], {n, 1, nmax}];
%t A005627 Table[a[n], {n, 0, nmax}] (* _Jean-François Alcover_, Jul 07 2024, after Maple code *)
%Y A005627 Cf. A000625.
%K A005627 nonn
%O A005627 0,4
%A A005627 _N. J. A. Sloane_
%E A005627 More terms from _Emeric Deutsch_, May 16 2004