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A000912 Expansion of (sqrt(1-4x^2) - sqrt(1-4x))/(2x). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Number of bond-rooted polyenoids with 2n-1 edges.
Partial sums are A129366.
REFERENCES
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
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. [Annotated scanned copy]
FORMULA
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
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A079995 A279322 A152011 * A228477 A360182 A169982
KEYWORD
nonn
AUTHOR
E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)
EXTENSIONS
More terms from Emeric Deutsch, Dec 19 2004
Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar
STATUS
approved

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Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)