A000912 -id:A000912 - cp's OEIS frontend

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

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

nonn

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]

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

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

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 *)

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

More terms from Sean A. Irvine, Oct 05 2015

A187916 a(n) = C(n) if n is odd, else C(n) - C(n/2); C(n) are Catalan numbers.

Original entry on oeis.org

1, 1, 5, 12, 42, 127, 429, 1416, 4862, 16754, 58786, 207880, 742900, 2674011, 9694845, 35356240, 129644790, 477633838, 1767263190, 6564103624, 24466267020, 91482504854, 343059613650, 1289903939312, 4861946401452, 18367352329252, 69533550916004

Offset: 1

Gary W. Adamson, Mar 15 2011

			a(6) = 127 = A000108(6) - A000108(3) = (132 - 5)
a(5) = 42 = A000108(5)

Cf. A000912, A000108

A187916 := proc(n)
    if type(n,'odd') then
        A000108(n);
    else
        A000108(n)-A000108(n/2);
    end if;
end proc:
seq(A187916(n),n=1..30) ; # R. J. Mathar, Jul 21 2015

Table[If[OddQ[n],CatalanNumber[n],CatalanNumber[n]-CatalanNumber[n/2]],{n,30}] (* Harvey P. Dale, May 22 2014 *)

Conjecture: -(n-2)*(n+2)*(n+1)*a(n) +8*(n+1)*(n^2-3*n+1)*a(n-1) +4*(-3*n^3+18*n^2-22*n+2)*a(n-2) -32*(n-2)*(n^2-3*n+1)*a(n-3) +16*

(2*n-7)*(n-3)*(2*n-3)*a(n-4)=0. - R. J. Mathar, Jul 21 2015

cp's OEIS Frontend

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

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