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A089022
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Number of walks of length 2n on the 3-regular tree beginning and ending at some fixed vertex.
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9
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1, 3, 15, 87, 543, 3543, 23823, 163719, 1143999, 8099511, 57959535, 418441191, 3043608351, 22280372247, 164008329423, 1213166815047, 9012417249663, 67208553680247, 502920171632943, 3775020828459687, 28415858155984863, 214444848602732247, 1622146752543427983
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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)
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) 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)
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI) x='x+O('x^66); Vec(4/(1+3*sqrt(1-8*x))) \\ Joerg Arndt, May 10 2013
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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