A000908 Atom-rooted polyenoids with n edges with symmetry class C_s.

0, 0, 1, 4, 14, 47, 164, 565, 1982, 6977, 24850, 89082, 321855, 1169853, 4276923, 15713799, 57998270, 214934984, 799473752, 2983682702, 11169374372, 41929478873, 157807392886, 595340271682, 2250901007539, 8527699269192, 32369066434276

Offset

0,4

Links

Table of n, a(n) for n=0..26.

B. N. Cyvin, E. Brendsdal, J. Brunvoll and S. J. Cyvin, Isomers of polyenes attached to benzene, Croatica Chemica Acta, 68 (1995), 63-73, C(x).

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

Formula

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

Maple

U0 := (1-sqrt(1-4*x))/2/x ;
V0 := 1+x*subs(x=x^2, U0) ;
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)
taylor(%, x=0, 60) ;
L := gfun[seriestolist](%) ;
seq(op(2*i+1, L), i=0..(nops(L)-1)/2) ; # R. J. Mathar, Jul 26 2019

Mathematica

u0[x_] := (1 - Sqrt[1 - 4 x])/(2 x); v0[x_] := 1 + x u0[x^2];
gf = Simplify[(u0[x]^3 - 3 u0[x^2] v0[x] - u0[x^3] + 3 v0[x^3])/6]
CoefficientList[gf + O[x]^30, x] (* Andrey Zabolotskiy, Feb 08 2023 *)

Crossrefs

Cf. A000912, A000913, A000935, A000936, A000941, A000942, A000947, A000948, A000953, A003446, A063786.

Sequence in context: A137284 A365987 A228178 * A121095 A264816 A332610

Adjacent sequences: A000905 A000906 A000907 * A000909 A000910 A000911

Keyword

nonn

Author

E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)

Extensions

More terms from Sean A. Irvine, Oct 05 2015

Status

approved