OFFSET
1,1
COMMENTS
Number of walks of length 2n+9 in the path graph P_10 from one end to the other one. - Emeric Deutsch, Apr 02 2004
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; 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 (9,-28,35,-15,1).
FORMULA
From Emeric Deutsch, Apr 02 2004: (Start)
G.f.: 1/(1 - 9*x + 28*x^2 - 35*x^3 + 15*x^4 - x^5) - 1.
a(n) = 9*a(n-1) - 28*a(n-2) + 35*a(n-3) - 15*a(n-4) + a(n-5). (End)
a(k) = Sum_{j=-infinity..infinity} (binomial(9+2*k, 11j+k-2) - binomial(9+2*k, 11j+k-1)) (a finite sum).
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); # Simon Plouffe in his 1992 dissertation
MATHEMATICA
LinearRecurrence[{9, -28, 35, -15, 1}, {9, 53, 260, 1156, 4845}, 25] (* Vincenzo Librandi, Jun 20 2017 *)
PROG
(Magma) I:=[9, 53, 260, 1156, 4845]; [n le 5 select I[n] else 9*Self(n-1)-28*Self(n-2)+35*Self(n-3)-15*Self(n-4)+Self(n-5): n in [1..30]]; // Vincenzo Librandi, Jun 20 2017
CROSSREFS
KEYWORD
nonn,walk,easy
AUTHOR
STATUS
approved