A045890 Catafusenes (see reference for precise definition).

1, 3, 12, 49, 204, 864, 3714, 16170, 71178, 316303, 1417248, 6396273, 29051856, 132700725, 609200640, 2809373915, 13008512040, 60457182345, 281919911460, 1318666411635, 6185356518660, 29088241615910, 137121834221346, 647821223533044, 3066862717614234, 14546629647573969

Offset

0,2

Comments

3-fold convolution of A002212. - Emeric Deutsch, Mar 13 2004

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

S. J. Cyvin et al., Enumeration and classification of certain polygonal systems representing polycyclic conjugated hydrocarbons: annelated catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.

Formula

G.f.: (1 - x - sqrt(1-6*x+5*x^2))^3/(8*x^3). - Emeric Deutsch, Mar 13 2004

a(n) = (3/n)*Sum_{j=1..n} binomial(n, j)*binomial(2j+2, j-1) for n >= 1. - Emeric Deutsch, Mar 25 2004

Recurrence: (n+2)*(n+3)*a(n) = 2*(n+2)*(3*n+4)*a(n-1) - 5*(n-2)*(n+3)*a(n-2). - Vaclav Kotesovec, Oct 08 2012

a(n) ~ 6*5^(n+1/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012

Maple

a := n->(3/n)*sum(binomial(n, j)*binomial(2*j+2, j-1), j=1..n): 1, seq(a(n), n=1..22);

Mathematica

a[n_] := 3*(Hypergeometric2F1[5/2, 1-n, 5, -4] + (n-1)*Hypergeometric2F1[7/2, 2-n, 6, -4]); a[0]=1; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Jun 13 2012, after Emeric Deutsch *)
CoefficientList[Series[(1-x-Sqrt[1-6x+5x^2])^3/(8x^3), {x, 0, 30}], x] (* Harvey P. Dale, Feb 07 2015 *)

Prog

(PARI) x='x+O('x^66); Vec((1-x-sqrt(1-6*x+5*x^2))^3/(8*x^3)) \\ Joerg Arndt, May 04 2013

Crossrefs

Cf. A002212.

Sequence in context: A037646 A012772 A012864 * A049673 A052703 A151170

Adjacent sequences: A045887 A045888 A045889 * A045891 A045892 A045893

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Emeric Deutsch, Mar 13 2004

Status

approved