cp's OEIS Frontend

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

%I A005023 M4409 #60 Jul 08 2025 16:27:45
%S A005023 7,34,143,560,2108,7752,28101,100947,360526,1282735,4552624,16131656,
%T A005023 57099056,201962057,714012495,2523515514,8916942687,31504028992,
%U A005023 111295205284,393151913464,1388758662221,4905479957435,17327203698086,61202661233823,216176614077600
%N A005023 Number of walks of length 2n+7 in the path graph P_8 from one end to the other.
%D A005023 W. Feller, An Introduction to Probability Theory and its Applications, 3rd ed, Wiley, New York, 1968, p. 96.
%D A005023 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005023 Vincenzo Librandi, <a href="/A005023/b005023.txt">Table of n, a(n) for n = 1..1000</a>
%H A005023 C. J. Everett and 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 A005023 C. J. Everett and 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 A005023 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 A005023 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A005023 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (7,-15,10,-1).
%F A005023 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
%F A005023 a(k) = sum(binomial(7+2k, 9j+k-2)-binomial(7+2k, 9j+k-1), j=-infinity..infinity) (a finite sum).
%p A005023 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);
%p A005023 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.
%t A005023 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 *)
%t A005023 LinearRecurrence[{7,-15,10,-1},{7,34,143,560},40] (* _Harvey P. Dale_, May 26 2013 *)
%t A005023 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 *)
%o A005023 (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
%Y A005023 Cf. A094829 (first differences), A094256 (essentially the same).
%K A005023 nonn,easy,walk
%O A005023 1,1
%A A005023 _N. J. A. Sloane_
%E A005023 Better definition from _Emeric Deutsch_, Apr 02 2004