|
%I M4526 #50 Sep 08 2022 08:44:33
%S 8,43,196,820,3264,12597,47652,177859,657800,2417416,8844448,32256553,
%T 117378336,426440955,1547491404,5610955132,20332248992,73645557469,
%U 266668876540,965384509651,3494279574288,12646311635088,45764967830976
%N Number of walks of length 2n+8 in the path graph P_9 from one end to the other.
%D W. Feller, An Introduction to Probability Theory and its Applications, 3rd ed, Wiley, New York, 1968, p. 96.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A005024/b005024.txt">Table of n, a(n) for n = 1..1000</a>
%H C. J. Everett, P. R. Stein, <a href="http://dx.doi.org/10.1016/0012-365X(77)90019-X">The combinatorics of random walk with absorbing barriers</a>, Discrete Math. 17 (1977), no. 1, 27-45.
%H C. J. Everett, P. R. Stein, <a href="/A005021/a005021.pdf">The combinatorics of random walk with absorbing barriers</a>, Discrete Math. 17 (1977), no. 1, 27-45. [Annotated scanned copy]
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-21,20,-5).
%F From _Emeric Deutsch_, Apr 02 2004: (Start)
%F G.f. (assuming a(0)=1): 1/(1 - 8x + 21x^2 - 20x^3 + 5x^4) - 1.
%F a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4). (End)
%F a(k) = sum(binomial(8+2k, 10j+k-2)-binomial(8+2k, 10j+k-1), j=-infinity..infinity) (a finite sum).
%p 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);
%p 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
%t 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 *)
%t 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 *)
%o (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
%o (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
%Y Cf. A005023. Truncated version of A094865.
%K nonn,easy,walk
%O 1,1
%A _N. J. A. Sloane_
%E Better definition from _Emeric Deutsch_, Apr 02 2004
|
