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A003241
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Number of achiral rooted trees.
(Formerly M1101)
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2
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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
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OFFSET
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1,3
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COMMENTS
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There may be an error in eq (37) in the Harary-Robinson paper. - R. J. Mathar, Sep 28 2011
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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MAPLE
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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) ;
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MATHEMATICA
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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]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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