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A005025
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Random walks.
(Formerly M4635)
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3
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9, 53, 260, 1156, 4845, 19551, 76912, 297275, 1134705, 4292145, 16128061, 60304951, 224660626, 834641671, 3094322026, 11453607152, 42344301686, 156404021389, 577291806894, 2129654436910, 7853149169635, 28949515515376
(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+9 in the path graph P_10 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/(1-9x+28x^2-35x^3+15x^4-x^5) - 1. a(n)=9a(n-1)-28a(n-2)+35a(n-3)-15a(n-4)+a(n-5). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 02 2004
a(k)=sum(binomial(9+2k, 11j+k-2)-binomial(9+2k, 11j+k-1), j=-infinity..infinity) (a finite sum).
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MAPLE
| a:=k->sum(binomial(9+2*k, 11*j+k-2), j=ceil((2-k)/11)..floor((11+k)/11))-sum(binomial(9+2*k, 11*j+k-1), j=ceil((1-k)/11)..floor((10+k)/11)): seq(a(k), k=1..28);
A005025:=-(9-28*z+35*z**2-15*z**3+z**4)/(-1+9*z-28*z**2+35*z**3-15*z**4+z**5); [Conjectured by S. Plouffe in his 1992 dissertation.]
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CROSSREFS
| Sequence in context: A055854 A202514 A122588 * A038761 A003698 A001688
Adjacent sequences: A005022 A005023 A005024 * A005026 A005027 A005028
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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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