This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A005025 M4635 #41 Sep 08 2022 08:44:33
%S A005025 9,53,260,1156,4845,19551,76912,297275,1134705,4292145,16128061,
%T A005025 60304951,224660626,834641671,3094322026,11453607152,42344301686,
%U A005025 156404021389,577291806894,2129654436910,7853149169635,28949515515376,106692395098433,393137817645838
%N A005025 Random walks.
%C A005025 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
%D A005025 W. Feller, An Introduction to Probability Theory and its Applications, 3rd ed, Wiley, New York, 1968, p. 96
%D A005025 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005025 Vincenzo Librandi, <a href="/A005025/b005025.txt">Table of n, a(n) for n = 1..1000</a>
%H A005025 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 A005025 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 A005025 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 A005025 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A005025 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (9,-28,35,-15,1).
%F A005025 From _Emeric Deutsch_, Apr 02 2004: (Start)
%F A005025 G.f.: 1/(1 - 9*x + 28*x^2 - 35*x^3 + 15*x^4 - x^5) - 1.
%F A005025 a(n) = 9*a(n-1) - 28*a(n-2) + 35*a(n-3) - 15*a(n-4) + a(n-5). (End)
%F A005025 a(k) = Sum_{j=-infinity..infinity} (binomial(9+2*k, 11j+k-2) - binomial(9+2*k, 11j+k-1)) (a finite sum).
%p A005025 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);
%p A005025 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
%t A005025 LinearRecurrence[{9, -28, 35, -15, 1}, {9, 53, 260, 1156, 4845}, 25] (* _Vincenzo Librandi_, Jun 20 2017 *)
%o A005025 (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
%K A005025 nonn,walk,easy
%O A005025 1,1
%A A005025 _N. J. A. Sloane_