A005024 Number of walks of length 2n+8 in the path graph P_9 from one end to the other. (Formerly M4526)

8, 43, 196, 820, 3264, 12597, 47652, 177859, 657800, 2417416, 8844448, 32256553, 117378336, 426440955, 1547491404, 5610955132, 20332248992, 73645557469, 266668876540, 965384509651, 3494279574288, 12646311635088, 45764967830976

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; 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 (8,-21,20,-5).

Formula

From Emeric Deutsch, Apr 02 2004: (Start)

G.f. (assuming a(0)=1): 1/(1 - 8x + 21x^2 - 20x^3 + 5x^4) - 1.

a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4). (End)

a(k) = sum(binomial(8+2k, 10j+k-2)-binomial(8+2k, 10j+k-1), j=-infinity..infinity) (a finite sum).

Maple

a:=k->sum(binomial(8+2*k, 10*j+k-2), j=ceil((2-k)/10)..floor((10+k)/10))-sum(binomial(8+2*k, 10*j+k-1), j=ceil((1-k)/10)..floor((9+k)/10)): seq(a(k), k=1..28);
A005024:=-(-8+21*z-20*z**2+5*z**3)/(5*z**2-5*z+1)/(z**2-3*z+1); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

CoefficientList[Series[(-5 z^3 + 20 z^2 - 21 z + 8)/((z^2 - 3 z + 1) (5 z^2 - 5 z + 1)), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 27 2011 *)
CoefficientList[Series[(1 / x) (1 / (1 - 8 x + 21 x^2 - 20 x^3 + 5 x^4) - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 08 2013 *)

Prog

(PARI) x='x+O('x^66); Vec(-1+1/((1-3*x+x^2)*(1-5*x+5*x^2))) \\ Joerg Arndt, May 01 2013
(Magma) I:=[8, 43, 196, 820]; [n le 4 select I[n] else 8*Self(n-1)-21*Self(n-2)+20*Self(n-3)-5*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 08 2013

Crossrefs

Cf. A005023. Truncated version of A094865.

Sequence in context: A000429 A055853 A137748 * A094865 A122880 A171479

Adjacent sequences: A005021 A005022 A005023 * A005025 A005026 A005027

Keyword

nonn,easy,walk

Author

N. J. A. Sloane

Extensions

Better definition from Emeric Deutsch, Apr 02 2004

Status

approved