A122737 Expansion of 1 - 3*x - sqrt(1 - 6*x + 5*x^2).

0, 2, 6, 20, 72, 274, 1086, 4438, 18570, 79174, 342738, 1502472, 6656436, 29756910, 134061570, 608072340, 2774495160, 12726088630, 58646299650, 271401086380, 1260750482760, 5876782098790, 27479558368170, 128861594138750, 605869334122602, 2855527261156394, 13488568550452446

Offset

1,2

Comments

Numbers of perifusenes with one internal vertex (see Cyvin et al. for precise definition).

For n>=2, a(n) is also the number of bi-wall directed polygons with perimeter 2n+2. Let us denote unit steps as follows: W=(-1,0), E=(1,0), N=(0,1), S=(0,-1). A bi-wall directed polygon is a self-avoiding polygon which can be factored as uv, where (1) u is a path which starts with an N step, ends with an S step, and can make N, E and S steps, and (2) v is a path which starts with a W step, ends with a W step, and can make W, S and E steps.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Jean-Luc Baril, José L. Ramírez, and Lina M. Simbaqueba, Counting prefixes of skew Dyck paths, J. Int. Seq., Vol. 24 (2021), Article 21.8.2.

S. J. Cyvin, F. Zhang and J. Brunvoll, Enumeration of perifusenes with one internal vertex: A complete mathematical solution, J. Math. Chem., 11 (1992), 283-292.

S. Feretic, Generating functions for bi-wall directed polygons, in: Proc. of the Seventh Int. Conf. on Lattice Path Combinatorics and Applications (eds. S. Rinaldi and S. G. Mohanty), Siena, 2010, 147-151.

Formula

For n>=1, a(n+1) = (3^(n+1)/(n*2^n))*Sum_{i=0..floor((n+1)/2)} ((-5/9)^i*binomial(n,i)*binomial(2*n-2*i,n-1)).

G.f.: 1/x - 3 - (1-x)/x/G(0), where G(k) = 1 + 4*x*(4*k+1)/( (4*k+2)*(1-x) - 2*x*(1-x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 24 2013

G.f.: (1-3*x - (1-5*x)*G(0))/x, where G(k) = 1 + 4*x*(4*k+1)/( (4*k+2)*(1-x) - 2*x*(1-x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 25 2013

a(n) ~ 5^(n-1/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 29 2013

D-finite with recurrence: n*a(n) + 3*(-2*n+3)*a(n-1) + 5*(n-3)*a(n-2) = 0. - R. J. Mathar, Jan 23 2020

a(n) = 2*A002212(n-1), n>1. - R. J. Mathar, Jan 23 2020

Example

There exist a(4)=20 bi-wall directed polygons with perimeter 2*4 + 2 = 10.

Mathematica

CoefficientList[Series[1 - 3*x - Sqrt[1 - 6*x + 5*x^2], {x, 0, 50}], x] (* G. C. Greubel, Mar 19 2017 *)

Prog

(PARI) x='x+O('x^66); concat([0], Vec(1-3*x-sqrt(1-6*x+5*x^2))) \\ Joerg Arndt, May 27 2013

Crossrefs

Sequence in context: A186996 A186576 A272485 * A338184 A348351 A150134

Adjacent sequences: A122734 A122735 A122736 * A122738 A122739 A122740

Keyword

nonn

Author

N. J. A. Sloane, Sep 24 2006

Extensions

Terms a(8)-a(20), better title, and extended edits from Svjetlan Feretic, May 24 2013

Status

approved