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 A000913 Number of bond-rooted polyenoids with n edges. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS 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 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

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Last modified December 11 07:36 EST 2019. Contains 329914 sequences. (Running on oeis4.)