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 and Daniel Nagaj, The pair-flip model: a very entangled translationally invariant spin chain, arXiv:1805.07168 [quant-ph], 2018.
Pakawut Jiradilok and Supanat Kamtue, Transportation Distance between Probability Measures on the Infinite Regular Tree, arXiv:2107.09876 [math.CO], 2021.
Ed Pegg Jr., K-Cayley Trees
FORMULA
G.f.: 4/(1+3*sqrt(1-8*x)).
a(n) = 2^n * binomial(2*n,n) - 3^(n-1) * Sum_{j=0..n-1} (2/3)^j*binomial(n+j,n). - 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
D-finite with recurrence 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
KEYWORD
easy,nonn
AUTHOR
Paul Boddington, Nov 11 2003
STATUS
approved