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 A005023 Random walks. (Formerly M4409) 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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 REFERENCES 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). LINKS _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. FORMULA 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). MAPLE 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.] MATHEMATICA 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 *) CROSSREFS Sequence in context: A014915 A137747 A094256 * A094891 A192803 A052161 Adjacent sequences:  A005020 A005021 A005022 * A005024 A005025 A005026 KEYWORD nonn,walk AUTHOR STATUS approved

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Last modified May 25 07:20 EDT 2013. Contains 225646 sequences.