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A089164
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Number of steps in all Schroeder paths (i.e. consisting of steps U=(1,1), D=(1,-1),H=(2,0) and never going below the x-axis) from (0,0) to (2n,0).
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0
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3, 19, 107, 591, 3259, 18019, 99987, 556831, 3111347, 17436915, 97981179, 551871087, 3114878571, 17613879747, 99768824355, 565962587199, 3214923140707, 18284737574611, 104110467624075, 593397580894351
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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FORMULA
| a(n)=(1/n)sum(k C(n, k-n)C(k, n-1), k=n..2n). G.f.=1/2-1/z+(2-7z+z^2)/[2z sqrt(1-6*z+z^2)].
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EXAMPLE
| a(2)=19 because the six Schroeder paths HH,HUD,UDH,UHD,UDUD,UUDD from (0,0) to (4,0) have 19 steps (i.e. letters) alltogether.
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MATHEMATICA
| f[n_] := Sum[k* Binomial[n, k - n] Binomial[k, n - 1], {k, n, 2 n}] /n; Array[f, 20] (* Or *)
Rest@ CoefficientList[ Series[(x - 2 + (2 - 7 x + x^2)/(Sqrt[1 - 6 x + x^2]))/(2 x), {x, 0, 20}], x] (* Robert G. Wilson v, Sep 12 2011 *)
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CROSSREFS
| Cf. A006318.
Sequence in context: A047029 A095120 A151539 * A072950 A130425 A103005
Adjacent sequences: A089161 A089162 A089163 * A089165 A089166 A089167
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 06 2003
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