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A005023 Number of walks of length 2n+7 in the path graph P_8 from one end to the other.
(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

REFERENCES

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

C. J. Everett, P. R. Stein, The combinatorics of random walk with absorbing barriers, Discrete Math. 17 (1977), no. 1, 27-45.

C. J. Everett, P. R. Stein, The combinatorics of random walk with absorbing barriers, Discrete Math. 17 (1977), no. 1, 27-45. [Annotated scanned copy]

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for linear recurrences with constant coefficients, signature (7,-15,10,-1).

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] (* Vladimir Joseph Stephan Orlovsky, Jun 27 2011 *)

LinearRecurrence[{7, -15, 10, -1}, {7, 34, 143, 560}, 40] (* Harvey P. Dale, May 26 2013 *)

CoefficientList[Series[(1 / x) (1 / (1 - 7 x + 15 x^2 - 10 x^3 + x^4) - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 08 2013 *)

PROG

(MAGMA)  I:=[7, 34, 143, 560]; [n le 4 select I[n] else 7*Self(n-1)-15*Self(n-2)+10*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 08 2013

CROSSREFS

Sequence in context: A014915 A137747 A273722 * A094256 A094891 A306376

Adjacent sequences:  A005020 A005021 A005022 * A005024 A005025 A005026

KEYWORD

nonn,walk

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Better definition from Emeric Deutsch, Apr 02 2004

STATUS

approved

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Last modified July 17 18:44 EDT 2019. Contains 325109 sequences. (Running on oeis4.)