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A003241 Number of achiral rooted trees.

%I A003241 M1101 #32 Apr 07 2020 11:07:27
%S A003241 1,1,2,4,8,15,26,45,71,110,168,247,351,503,700,944,1294,1719,2267,
%T A003241 2961,3839,4891,6297,7891,9912,12347,15381,18784,23203,28138,34233,
%U A003241 41275,49824,59306,71309,84268,100127,118045,139472,162659
%N A003241 Number of achiral rooted trees.
%C A003241 There may be an error in eq (37) in the Harary-Robinson paper. - _R. J. Mathar_, Sep 28 2011
%D A003241 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003241 Jean-François Alcover, <a href="/A003241/b003241.txt">Table of n, a(n) for n = 1..80</a>
%H A003241 F. Harary and R. W. Robinson, <a href="http://dx.doi.org/10.1515/crll.1975.278-279.322">The number of achiral trees</a>, J. Reine Angew. Math., 278 (1975), 322-335.
%H A003241 F. Harary and R. W. Robinson, <a href="/A002995/a002995_1.pdf">The number of achiral trees</a>, J. Reine Angew. Math., 278 (1975), 322-335. (Annotated scanned copy)
%H A003241 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H A003241 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%p A003241 L := BFILETOLIST("b003238.txt") ;
%p A003241 Pofxn := proc(n)
%p A003241         global L;
%p A003241         add( op(i,L)*x^(i+1),i=1..120) ;
%p A003241         subs(x=x^n,%) ;
%p A003241 end proc:
%p A003241 P := Pofxn(1) ;
%p A003241 Rn := proc(n)
%p A003241         global L;
%p A003241         (Pofxn(n-2)*Pofxn(2)+Pofxn(n-1)*Pofxn(1)-Pofxn(n))/x^(n-1) ;
%p A003241 end proc:
%p A003241 Px2 := Pofxn(2) ;
%p A003241 Px3 := Pofxn(3) ;
%p A003241 Px4 := Pofxn(4) ;
%p A003241 # eq (37) seems not to work
%p A003241 # 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 ;
%p A003241 #use eqs (39)-(44) instead
%p A003241 R := x+P+(P^2+Px2)/2/x+P*Px2/x^2+P*Px3/x^3+(Px2^2-Px4)/2/x^3 :
%p A003241 # heuristics, adding up to R^(40) suffices for first 80 terms
%p A003241 for n from 5 to 40 do
%p A003241         R := R+Rn(n) :
%p A003241 end do:
%p A003241 taylor(R,x=0,80) ;
%p A003241 gfun[seriestolist](%) ; # _R. J. Mathar_, Sep 28 2011
%t A003241 L = Cases[Import["https://oeis.org/A003238/b003238.txt", "Table"], {_, _}][[All, 2]];
%t A003241 Pofxn[n_] := Sum[x^(i+1) L[[i]], {i, 1, 120}] /. x -> x^n;
%t A003241 P = Pofxn[1];
%t A003241 Rn[n_] := (1/x^(n-1))(Pofxn[2] Pofxn[n-2] + Pofxn[1] Pofxn[n-1] - Pofxn[n]);
%t A003241 Px2 = Pofxn[2]; Px3 = Pofxn[3]; Px4 = Pofxn[4];
%t A003241 R = (P^2 + Px2)/(2x) + (P Px2)/x^2 + (P Px3)/x^3 + P + (Px2^2 - Px4)/(2x^3) + x;
%t A003241 For[n = 5, n <= 40, n++, R += Rn[n]];
%t A003241 CoefficientList[R + O[x]^41, x] // Rest (* _Jean-François Alcover_, Apr 06 2020, from Maple *)
%K A003241 nonn
%O A003241 1,3
%A A003241 _N. J. A. Sloane_
%E A003241 Extended by _R. J. Mathar_, Sep 28 2011