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%I #24 Feb 16 2023 14:54:05
%S 0,0,1,4,14,47,164,565,1982,6977,24850,89082,321855,1169853,4276923,
%T 15713799,57998270,214934984,799473752,2983682702,11169374372,
%U 41929478873,157807392886,595340271682,2250901007539,8527699269192,32369066434276
%N Atom-rooted polyenoids with n edges with symmetry class C_s.
%H B. N. Cyvin, E. Brendsdal, J. Brunvoll and S. J. Cyvin, <a href="https://hrcak.srce.hr/176546">Isomers of polyenes attached to benzene</a>, Croatica Chemica Acta, 68 (1995), 63-73, C(x).
%H S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, <a href="http://dx.doi.org/10.1021/ci00026a012">Enumeration of polyene hydrocarbons: a complete mathematical solution</a>, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
%H S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, <a href="/A002057/a002057.pdf">Enumeration of polyene hydrocarbons: a complete mathematical solution</a>, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]
%F a(n) = A003446(n+1) - u((n-3)/6) - (u(n/3) - u((n-3)/6))/2 - (u(n/2) + (u((n+1)/2) - u((n-3)/6))) for n > 0 where u(n) = binomial(2*n, n)/(n+1) if n is an integer and 0 otherwise. - _Sean A. Irvine_, Oct 05 2015
%p U0 := (1-sqrt(1-4*x))/2/x ;
%p V0 := 1+x*subs(x=x^2,U0) ;
%p C := ( subs(x=x^2,U0)^3 -3*subs(x=x^4,U0)*subs(x=x^2,V0) -subs(x=x^6,U0) +3*subs(x=x^6,V0) )/6 ; # (19)
%p taylor(%,x=0,60) ;
%p L := gfun[seriestolist](%) ;
%p seq(op(2*i+1,L),i=0..(nops(L)-1)/2) ; # _R. J. Mathar_, Jul 26 2019
%t u0[x_] := (1 - Sqrt[1 - 4 x])/(2 x); v0[x_] := 1 + x u0[x^2];
%t gf = Simplify[(u0[x]^3 - 3 u0[x^2] v0[x] - u0[x^3] + 3 v0[x^3])/6]
%t CoefficientList[gf + O[x]^30, x] (* _Andrey Zabolotskiy_, Feb 08 2023 *)
%Y Cf. A000912, A000913, A000935, A000936, A000941, A000942, A000947, A000948, A000953, A003446, A063786.
%K nonn
%O 0,4
%A E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)
%E More terms from _Sean A. Irvine_, Oct 05 2015
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