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A005023
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Random walks.
(Formerly M4409)
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4
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7, 34, 143, 560, 2108, 7752, 28101, 100947, 360526, 1282735, 4552624, 16131656, 57099056, 201962057, 714012495, 2523515514, 8916942687, 31504028992, 111295205284, 393151913464, 1388758662221, 4905479957435, 17327203698086, 61202661233823, 216176614077600
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OFFSET
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1,1
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COMMENTS
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Number of walks of length 2n+7 in the path graph P_8 from one end to the other one. - Emeric Deutsch, Apr 02 2004
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REFERENCES
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Everett, C. J.; Stein, P. R.; The combinatorics of random walk with absorbing barriers. Discrete Math. 17 (1977), no. 1, 27-45.
W. Feller, An Introduction to Probability Theory and its Applications, 3rd ed, Wiley, New York, 1968, p. 96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Table of n, a(n) for n=1..25.
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
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G.f.=1/(1-7x+15x^2-10x^3+x^4) - 1. a(n)=7a(n-1)-15a(n-2)+10a(n-3)-a(n-4). - Emeric Deutsch, Apr 02 2004
a(k)=sum(binomial(7+2k, 9j+k-2)-binomial(7+2k, 9j+k-1), j=-infinity..infinity) (a finite sum).
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MAPLE
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a:=k->sum(binomial(7+2*k, 9*j+k-2), j=ceil((2-k)/9)..floor((9+k)/9))-sum(binomial(7+2*k, 9*j+k-1), j=ceil((1-k)/9)..floor((8+k)/9)): seq(a(k), k=1..28);
A005023:=-(-7+15*z-10*z**2+z**3)/(z-1)/(z**3-9*z**2+6*z-1); [Conjectured by Simon Plouffe in his 1992 dissertation.]
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MATHEMATICA
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CoefficientList[Series[(-z^3 + 10 z^2 - 15 z + 7)/(z^4 - 10 z^3 + 15 z^2 - 7 z + 1), {z, 0, 100}], z] (* From Vladimir Joseph Stephan Orlovsky, Jun 27 2011 *)
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CROSSREFS
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Sequence in context: A014915 A137747 A094256 * A094891 A192803 A052161
Adjacent sequences: A005020 A005021 A005022 * A005024 A005025 A005026
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KEYWORD
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nonn,walk
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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