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%I #32 Dec 10 2023 11:10:38
%S 0,1,2,12,38,143,490,1768,6268,22610,81620,297160,1086172,3991995,
%T 14731290,54587280,202992808,757398510,2834493948,10637507400,
%U 40023577524,150946230006,570534370692,2160865067312,8199710635816
%N Number of bond-rooted polyenoids with n edges.
%H S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin, and E. K. Lloyd, <a href="https://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 From _Emeric Deutsch_, Dec 19 2004: (Start)
%F 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.
%F 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)
%p 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
%t 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_ *)
%o (PARI) c(n) = if ((n<0) || (denominator(n) > 1), 0, binomial(2*n,n)/(n+1));
%o 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
%Y Cf. A000108 (Catalan).
%K nonn
%O 1,3
%A E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)
%E More terms from _Emeric Deutsch_, Dec 19 2004
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