A003241 Number of achiral rooted trees. (Formerly M1101)

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

1,3

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

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 *)

Crossrefs

Sequence in context: A187154 A179001 A222147 * A279320 A182844 A191630

Adjacent sequences: A003238 A003239 A003240 * A003242 A003243 A003244

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Extended by R. J. Mathar, Sep 28 2011

Status

approved