A005627 - OEIS

%I M0698 #42 Jul 07 2024 03:34:17

%S 1,1,1,2,3,5,8,14,23,41,69,122,208,370,636,1134,1963,3505,6099,10908,

%T 19059,34129,59836,107256,188576,338322,596252,1070534,1890548,

%U 3396570,6008908,10801816,19139155,34422537,61074583,109894294,195217253

%N Number of achiral planted trees with n nodes.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H 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 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 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>.

%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>.

%F 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 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 nmax = 36;

%t s[0] = s[1] = 1; s[_] = 0;

%t 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 a[0] = a[1] = 1;

%t Do[a[n+1] = Sum[s[k]*a[n-2*k], {k, 0, Floor[n/2]}], {n, 1, nmax}];

%t Table[a[n], {n, 0, nmax}] (* _Jean-François Alcover_, Jul 07 2024, after Maple code *)

%Y Cf. A000625.

%K nonn

%O 0,4

%A _N. J. A. Sloane_

%E More terms from _Emeric Deutsch_, May 16 2004