A000913 Number of bond-rooted polyenoids with n edges.

0, 1, 2, 12, 38, 143, 490, 1768, 6268, 22610, 81620, 297160, 1086172, 3991995, 14731290, 54587280, 202992808, 757398510, 2834493948, 10637507400, 40023577524, 150946230006, 570534370692, 2160865067312, 8199710635816

Offset

1,3

Links

Table of n, a(n) for n=1..25.

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

From Emeric Deutsch, Dec 19 2004: (Start)

a(n) = (1/4)*c(n+2) - (1/2)*c(n+1) - (3/4)*c((n+1)/2) + (1/2)*c((n-1)/4), where c(n) = binomial(2n, n)/(n+1) are the Catalan numbers for n a nonnegative integer and 0 otherwise.

G.f.: (-4x + 8x^2 - sqrt(1-4x) + 2x*sqrt(1-4x) + 3*sqrt(1-4x^2) - 2*sqrt(1-4x^4))/(8x^3). (End)

Maple

c:=proc(n) if floor(n)=n then binomial(2*n, n)/(n+1) else 0 fi end:a:=n->(1/4)*c(n+2)-(1/2)*c(n+1)-(3/4)*c((n+1)/2)+(1/2)*c((n-1)/4):seq(a(n), n=1..27); # Emeric Deutsch, Dec 19 2004

Mathematica

c[n_] := If[Floor[n] == n, Binomial[2 n, n]/(n + 1), 0]; a[n_] := (1/4)*c[n + 2] - (1/2)*c[n + 1] - (3/4)*c[(n + 1)/2] + (1/2)*c[(n - 1)/4]; Table[a[n], {n, 1, 25}] (* James C. McMahon, Dec 09 2023 after MAPLE by Emeric Deutsch *)

Prog

(PARI) c(n) = if ((n<0) || (denominator(n) > 1), 0, binomial(2*n, n)/(n+1));
a(n) = (1/4)*c(n+2) - (1/2)*c(n+1) - (3/4)*c((n+1)/2) + (1/2)*c((n-1)/4); \\ Michel Marcus, Aug 26 2019

Crossrefs

Cf. A000108 (Catalan).

Sequence in context: A305864 A324027 A035597 * A026575 A048349 A009632

Adjacent sequences: A000910 A000911 A000912 * A000914 A000915 A000916

Keyword

nonn

Author

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

Extensions

More terms from Emeric Deutsch, Dec 19 2004

Status

approved