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 and P. R. Stein, The combinatorics of random walk with absorbing barriers, Discrete Math. 17 (1977), no. 1, 27-45.
C. J. Everett and 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; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 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
KEYWORD
nonn,easy,walk
AUTHOR
EXTENSIONS
Better definition from Emeric Deutsch, Apr 02 2004
STATUS
approved