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A005021
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Random walks (binomial transform of A006054).
(Formerly M3888)
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7
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1, 5, 19, 66, 221, 728, 2380, 7753, 25213, 81927, 266110, 864201, 2806272, 9112264, 29587889, 96072133, 311945595, 1012883066, 3288813893, 10678716664, 34673583028, 112584429049, 365559363741, 1186963827439, 3854047383798
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Number of walks of length 2n+5 in the path graph P_6 from one end to the other one. Example: a(1)=5 because in the path ABCDEF we have ABABCDEF, ABCBCDEF, ABCDCDEF, ABCDEDEF and ABCDEFEF. - 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.
N. J. A. Sloane, Transforms
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FORMULA
| G.f.: 1/(1-5x+6x^2-x^3). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 02 2004
a(n)=5a(n-1)-6a(n-2)+a(n-3). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 02 2004
a(n)=sum(binomial(5+2k, 7j+k-2)-binomial(5+2k, 7j+k-1), j=-infinity..infinity) (a finite sum).
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MAPLE
| a:=k->sum(binomial(5+2*k, 7*j+k-2), j=ceil((2-k)/7)..floor((7+k)/7))-sum(binomial(5+2*k, 7*j+k-1), j=ceil((1-k)/7)..floor((6+k)/7)): seq(a(k), k=0..25);
A005021:=-(z-1)*(z-5)/(-1+5*z-6*z**2+z**3); [Conjectured by S. Plouffe in his 1992 dissertation. Gives sequence apart from the initial 1.]
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CROSSREFS
| Double partial sums of A060557. Bisection of A052547.
Cf. A094789, A094790, A080937.
Sequence in context: A124806 A059509 A137745 * A067325 A121525 A163872
Adjacent sequences: A005018 A005019 A005020 * A005022 A005023 A005024
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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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