A000912 Expansion of (sqrt(1-4x^2) - sqrt(1-4x))/(2x).

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

Offset: 0

Author

E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)

Keywords

• nonn

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]

Programs

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 *)

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

D-finite with recurrence 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

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