%I A005025 M4635
%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
%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 (deutsch(AT)duke.poly.edu), Apr 02 
               2004
%D A005025 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005025 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques 
               Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%D A005025 Everett, C. J.; Stein, P. R.; The combinatorics of random walk with absorbing 
               barriers. Discrete Math. 17 (1977), no. 1, 27-45.
%D A005025 W. Feller, An Introduction to Probability Theory and its Applications, 
               3rd ed, Wiley, New York, 1968, p. 96
%H A005025 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005025 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A005025 G.f.=1/(1-9x+28x^2-35x^3+15x^4-x^5) - 1. a(n)=9a(n-1)-28a(n-2)+35a(n-3)-15a(n-4)+a(n-5). 
               - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 02 2004
%F A005025 a(k)=sum(binomial(9+2k, 11j+k-2)-binomial(9+2k, 11j+k-1), j=-infinity..infinity) 
               (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); 
               [Conjectured by S. Plouffe in his 1992 dissertation.]
%Y A005025 Sequence in context: A126085 A055854 A122588 this_sequence A038761 A003698 
               A001688
%Y A005025 Adjacent sequences: A005022 A005023 A005024 this_sequence A005026 A005027 
               A005028
%K A005025 nonn,walk
%O A005025 1,1
%A A005025 N. J. A. Sloane (njas(AT)research.att.com).