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Search: id:A089022
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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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7
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| 1, 3, 15, 87, 543, 3543, 23823, 163719, 1143999, 8099511, 57959535, 418441191, 3043608351, 22280372247, 164008329423, 1213166815047, 9012417249663, 67208553680247
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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.
Hankel transform is A133460 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 01 2007
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LINKS
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Ed Pegg Jr., K-Cayley Trees
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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( (2/3)^j*binomial(x+j,x), j=0..x-1) - Tobias Friedrich (tfried(AT)mpi-inf.mpg.de), Jun 12 2007
a(n) = Sum{k, 0<=k<=n}A039599(n,k)*2^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 25 2007
Contribution from Paul Barry (pbarry(AT)wit.ie), Sep 04 2009: (Start)
G.f.: 1/(1-3x/(1-2x/(1-2x/(1-2x/(1-2x/(1-... (continued fraction);
G.f.: (1-2xc(x))/(1-3x-2xc(x)), c(x) the g.f. of A000108. (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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CROSSREFS
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Sequence in context: A001931 A075841 A152596 this_sequence A132371 A127785 A074541
Adjacent sequences: A089019 A089020 A089021 this_sequence A089023 A089024 A089025
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Boddington (psb(AT)maths.warwick.ac.uk), Nov 11 2003
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