A003241 Number of achiral rooted trees.

1, 1, 2, 4, 8, 15, 26, 45, 71, 110, 168, 247, 351, 503, 700, 944, 1294, 1719, 2267, 2961, 3839, 4891, 6297, 7891, 9912, 12347, 15381, 18784, 23203, 28138, 34233, 41275, 49824, 59306, 71309, 84268, 100127, 118045, 139472, 162659

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

There may be an error in eq (37) in the Harary-Robinson paper. - R. J. Mathar, Sep 28 2011

References

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

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..80
• F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335.
• F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335. (Annotated scanned copy)
• Index entries for sequences related to rooted trees
• Index entries for sequences related to trees

Programs

Maple

L := BFILETOLIST("b003238.txt") ;
Pofxn := proc(n)
        global L;
        add( op(i,L)*x^(i+1),i=1..120) ;
        subs(x=x^n,%) ;
end proc:
P := Pofxn(1) ;
Rn := proc(n)
        global L;
        (Pofxn(n-2)*Pofxn(2)+Pofxn(n-1)*Pofxn(1)-Pofxn(n))/x^(n-1) ;
end proc:
Px2 := Pofxn(2) ;
Px3 := Pofxn(3) ;
Px4 := Pofxn(4) ;
# eq (37) seems not to work
# R := 2*x+P^2/x^2+(1-x)*P/x*(Px2/x^2-1)-(P^2-Px2)/2/x -Px3/x^2-(Px2^2-Px4)/2/x^3 ;
#use eqs (39)-(44) instead
R := x+P+(P^2+Px2)/2/x+P*Px2/x^2+P*Px3/x^3+(Px2^2-Px4)/2/x^3 :
# heuristics, adding up to R^(40) suffices for first 80 terms
for n from 5 to 40 do
        R := R+Rn(n) :
end do:
taylor(R,x=0,80) ;
gfun[seriestolist](%) ; # R. J. Mathar, Sep 28 2011

Mathematica

L = Cases[Import["https://oeis.org/A003238/b003238.txt", "Table"], {, }][[All, 2]];
Pofxn[n_] := Sum[x^(i+1) L[[i]], {i, 1, 120}] /. x -> x^n;
P = Pofxn[1];
Rn[n_] := (1/x^(n-1))(Pofxn[2] Pofxn[n-2] + Pofxn[1] Pofxn[n-1] - Pofxn[n]);
Px2 = Pofxn[2]; Px3 = Pofxn[3]; Px4 = Pofxn[4];
R = (P^2 + Px2)/(2x) + (P Px2)/x^2 + (P Px3)/x^3 + P + (Px2^2 - Px4)/(2x^3) + x;
For[n = 5, n <= 40, n++, R += Rn[n]];
CoefficientList[R + O[x]^41, x] // Rest (* Jean-François Alcover, Apr 06 2020, from Maple *)

Extensions

Extended by R. J. Mathar, Sep 28 2011