

A122737


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


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
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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 biwall 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 biwall directed polygon is a selfavoiding 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



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*n2*i,n1)).
G.f.: 1/x  3  (1x)/x/G(0), where G(k) = 1 + 4*x*(4*k+1)/( (4*k+2)*(1x)  2*x*(1x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1x)*(k+1)/G(k+1))); (continued fraction).  Sergei N. Gladkovskii, Jun 24 2013
G.f.: (13*x  (15*x)*G(0))/x, where G(k) = 1 + 4*x*(4*k+1)/( (4*k+2)*(1x)  2*x*(1x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1x)*(k+1)/G(k+1))); (continued fraction).  Sergei N. Gladkovskii, Jun 25 2013
Dfinite with recurrence: n*a(n) + 3*(2*n+3)*a(n1) + 5*(n3)*a(n2) = 0.  R. J. Mathar, Jan 23 2020


EXAMPLE

There exist a(4)=20 biwall 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(13*xsqrt(16*x+5*x^2))) \\ Joerg Arndt, May 27 2013


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS

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


STATUS

approved



