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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

T. D. Noe, Table of n, a(n) for n = 0..200

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 A169982 A243323

Adjacent sequences:  A000909 A000910 A000911 * A000913 A000914 A000915

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 January 20 00:34 EST 2022. Contains 350467 sequences. (Running on oeis4.)