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A000912
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Expansion of (sqrt(1-4x^2) - sqrt(1-4x))/(2x).
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3
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1, 0, 2, 4, 14, 40, 132, 424, 1430, 4848, 16796, 58744, 208012, 742768, 2674440, 9694416, 35357670, 129643360, 477638700, 1767258328, 6564120420, 24466250224, 91482563640, 343059554864, 1289904147324, 4861946193440, 18367353072152
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OFFSET
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0,3
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COMMENTS
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Number of bond-rooted polyenoids with 2n-1 edges.
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REFERENCES
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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
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LINKS
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FORMULA
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a(n) = C(n) if n is even and a(n) = C(n) -C((n-1)/2) if n is odd, where C(n) = binomial(2n, n)/(n+1) are the Catalan numbers (A000108). a(n) = 2*A000150(n) for n > 0. - Emeric Deutsch, Dec 19 2004
G.f.: c(x) - x*c(x^2), where c(x) = g.f. for A000108; a(n) = C(n) - C((n-1)/2)(1-(-1)^n)/2, C(n) = A000108(n). - Paul Barry, Apr 11 2007
Conjecture: n*(n+1)*a(n) - 6*n*(n-1)*a(n-1) + 4*(2*n^2-10*n+9)*a(n-2) + 8*(n^2+n-9)*a(n-3) - 48*(n-3)*(n-4)*a(n-4) + 32*(2*n-9)*(n-5)*a(n-5) = 0. - R. J. Mathar, Nov 24 2012
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MAPLE
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c:=n->binomial(2*n, n)/(n+1):a:=proc(n) if n mod 2 = 1 then c(n+1) else c(n+1)-c(n/2) fi end: seq(a(n), n=0..28); # Emeric Deutsch, Dec 19 2004
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MATHEMATICA
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nn = 200; CoefficientList[Series[(Sqrt[1 - 4 x^2] - Sqrt[1 - 4 x])/(2 x), {x, 0, nn}], x] (* T. D. Noe, Jun 20 2012 *)
Table[If[EvenQ[n], CatalanNumber[n], CatalanNumber[n]-CatalanNumber[(n-1)/ 2]], {n, 0, 30}] (* Harvey P. Dale, Oct 30 2013 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)
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EXTENSIONS
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STATUS
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approved
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