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A089022 Number of walks of length 2n on the 3-regular tree beginning and ending at some fixed vertex. 9
1, 3, 15, 87, 543, 3543, 23823, 163719, 1143999, 8099511, 57959535, 418441191, 3043608351, 22280372247, 164008329423, 1213166815047, 9012417249663, 67208553680247, 502920171632943, 3775020828459687, 28415858155984863, 214444848602732247, 1622146752543427983 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The generating function for the corresponding sequence for the m-regular tree is 2*(m-1)/(m-2+m*sqrt(1-4*(m-1)*x)). When m=2 this reduces to the usual generating function for the central binomial coefficients.

Hankel transform is A133460. - Philippe Deléham, Dec 01 2007

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Libor Caha, Daniel Nagaj, The pair-flip model: a very entangled translationally invariant spin chain, arXiv:1805.07168 [quant-ph], 2018.

Ed Pegg Jr., K-Cayley Trees

FORMULA

G.f.: 4/(1+3*sqrt(1-8*x)).

a(n) = 2^x * binomial(2*x,x) - 3^(x-1) * Sum_{j=0..x-1} (2/3)^j*binomial(x+j,x). - Tobias Friedrich (tfried(AT)mpi-inf.mpg.de), Jun 12 2007

a(n) = Sum_{k=0..n} A039599(n,k)*2^(n-k). - Philippe Deléham, Aug 25 2007

From Paul Barry, Sep 04 2009: (Start)

G.f.: 1/(1-3x/(1-2x/(1-2x/(1-2x/(1-2x/(1-... (continued fraction);

G.f.: (1-2*x*c(x))/(1-3*x-2*x*c(x)), where c(x) is the g.f. of A000108. (End)

a(n) = A126087(2n). - Philippe Deléham, Nov 02 2011

Conjecture: n*a(n) + (12-17*n)*a(n-1) + 36*(2n-3)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011

a(n) ~ 6*8^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012

From Karol A. Penson, Sep 06 2014: (Start)

a(n) is the (2*n)-th moment of a positive function W(x)=(3/Pi)*sqrt(8-x^2)/(9-x^2), on the segment x=(0,2*sqrt(2)), in Maple notation: a(n)=int(x^(2*n)*W(x),x=0..2*sqrt(2));

a(n) is the special value of hypergeometric function 2F1, in Maple notation: a(n)=2*8^n*GAMMA(n+1/2)*hypergeom([1,n+1/2],[n+2],8/9)/(3*sqrt(Pi)*(n+1)!). (End)

a(n) = A151374(n)*hypergeom([1,n+1/2],[n+2],8/9)*(2/3). - Peter Luschny, Sep 06 2014

EXAMPLE

a(2) = 15 because there are 3*3=9 walks whose second step is to return to the starting vertex and 3*2=6 walks whose second step is to move away from the starting vertex.

MAPLE

A000602 := x -> 2^x*binomial(2*x, x)-9^x+1/3*2^x*binomial(2*x, x) * hypergeom([1, 2*x+1], [x+1], 2/3); # Tobias Friedrich (tfried(AT)mpi-inf.mpg.de), Jun 12 2007

MATHEMATICA

Table[2^n*Binomial[2*n, n]-3^(n-1)*Sum[(2/3)^k*Binomial[n+k, n], {k, 0, n-1}], {n, 0, 20}] (* or *)

CoefficientList[Series[4/(1+3*Sqrt[1-8*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

PROG

(PARI) x='x+O('x^66); Vec(4/(1+3*sqrt(1-8*x))) \\ Joerg Arndt, May 10 2013

CROSSREFS

Column k=3 of A183135.

Sequence in context: A152596 A278392 A168503 * A246538 A132371 A192989

Adjacent sequences:  A089019 A089020 A089021 * A089023 A089024 A089025

KEYWORD

easy,nonn

AUTHOR

Paul Boddington, Nov 11 2003

STATUS

approved

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Last modified October 23 07:20 EDT 2018. Contains 316520 sequences. (Running on oeis4.)