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A122951
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Number of walks from (0,0) to (n,n) in the region x >= y with the steps (1,0), (0,1), (2,0) and (0,2).
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7
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1, 1, 5, 22, 117, 654, 3843, 23323, 145172, 921508, 5942737, 38825546, 256431172, 1709356836, 11485249995, 77703736926, 528893901963, 3619228605738, 24884558358426, 171828674445330, 1191050708958096, 8284698825305832
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| When this walk is further restricted to the subset of the plane x-y <= 2, this gives the sequence A046717. Similarly, the sequence for such a walk restricted to x-y <= w (w > 2) is not present in the OEIS. The reference provided proves recurrences for generating functions in w.
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LINKS
| Arvind Ayyer and Doron Zeilberger, The Number of [Old-Time] Basketball games with Final Score n:n where the Home Team was never losing but also never ahead by more than w Points
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FORMULA
| In Maple, GF is given by solve(z^4*F^4-2*z^3*F^3-z^2*F^3+2*z^2*F^2+3*z*F^2-2*z*F-F+1,F);
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EXAMPLE
| a(2)=5 because we can reach (2,2) in the following ways:
(0,0),(1,0),(1,1),(2,1),(2,2)
(0,0),(2,0),(2,2)
(0,0),(1,0),(2,0),(2,2)
(0,0),(2,0),(2,1),(2,2)
(0,0),(1,0),(2,0),(2,1),(2,2)
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MATHEMATICA
| f[x_] = (2x+Sqrt[4(x-2)x+1] - Sqrt[2]Sqrt[2x(-2x + Sqrt[4(x-2)x+1]-1) + Sqrt[4(x-2)x+1]+1]+1)/(4x^2);
CoefficientList[Series[f[x], {x, 0, 21}], x]
(* From J.F.Alcover, May 19 2011, after g.f. *)
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CROSSREFS
| Cf. A000108, A046717.
Sequence in context: A127618 A127619 A127620 * A184181 A020003 A131460
Adjacent sequences: A122948 A122949 A122950 * A122952 A122953 A122954
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KEYWORD
| nice,nonn
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AUTHOR
| Arvind Ayyer (ayyer(AT)physics.rutgers.edu), Oct 25 2006
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