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A005629 Number of achiral trees with n nodes.

%I A005629 M0677 #38 Apr 07 2025 12:00:24
%S A005629 1,1,1,2,3,5,7,14,21,40,61,118,186,355,567,1081,1755,3325,5454,10306,
%T A005629 17070,32136,53628,100704,169175,316874,535267,1000524,1698322,
%U A005629 3168500,5400908,10059823,17211368,32010736,54947147,102059572,175702378
%N A005629 Number of achiral trees with n nodes.
%D A005629 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005629 L. Bytautats and D. J. Klein, <a href="https://doi.org/10.1021/ci990021g">Alkane Isomer Combinatorics: Stereostructure enumeration and graph-invariant and molecular-property distributions</a>, J. Chem. Inf. Comput. Sci 39 (1999) 803, Table 1.
%H A005629 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 A005629 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 A005629 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A005629 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
%p A005629 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
%t A005629 nmax = 37;
%t A005629 s[0] = s[1] = 1; s[_] = 0;
%t A005629 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 A005629 p[0] = 1;
%t A005629 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}];
%t A005629 a[n_] := s[n] - p[n];
%t A005629 Table[a[n], {n, 1, nmax}] (* _Jean-François Alcover_, Jul 07 2024, after Maple code *)
%Y A005629 Cf. A000625, A005627.
%K A005629 nonn
%O A005629 1,4
%A A005629 _N. J. A. Sloane_
%E A005629 Corrected and extended by _Emeric Deutsch_, Nov 21 2004