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A005024
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Random walks.
(Formerly M4526)
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2
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8, 43, 196, 820, 3264, 12597, 47652, 177859, 657800, 2417416, 8844448, 32256553, 117378336, 426440955, 1547491404, 5610955132, 20332248992, 73645557469, 266668876540, 965384509651, 3494279574288, 12646311635088, 45764967830976
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Number of walks of length 2n+8 in the path graph P_9 from one end to the other one. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 02 2004
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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).
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LINKS
| S. 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.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
| G.f.=1/(1-8x+21x^2-20x^3+5x^4) - 1. a(n)=8a(n-1)-21a(n-2)+20a(n-3)-5a(n-4). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 02 2004
a(k)=sum(binomial(8+2k, 10j+k-2)-binomial(8+2k, 10j+k-1), j=-infinity..infinity) (a finite sum).
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MAPLE
| a:=k->sum(binomial(8+2*k, 10*j+k-2), j=ceil((2-k)/10)..floor((10+k)/10))-sum(binomial(8+2*k, 10*j+k-1), j=ceil((1-k)/10)..floor((9+k)/10)): seq(a(k), k=1..28);
A005024:=-(-8+21*z-20*z**2+5*z**3)/(5*z**2-5*z+1)/(z**2-3*z+1); [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| CoefficientList[Series[(-5 z^3 + 20 z^2 - 21 z + 8)/((z^2 - 3 z + 1) (5 z^2 - 5 z + 1)), {z, 0, 100}], z] (* From Vladimir Joseph Stephan Orlovsky, Jun 27 2011 *)
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CROSSREFS
| Cf. A005023.
Sequence in context: A000429 A055853 A137748 * A094865 A122880 A171479
Adjacent sequences: A005021 A005022 A005023 * A005025 A005026 A005027
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KEYWORD
| nonn,walk
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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