%I A092765
%S A092765 1,0,4,6,36,100,430,1470,5796,21336,82404,312180,1203246,4617756,
%T A092765 17846686,68974906,267498660,1038555024,4040525320,15739195680,
%U A092765 61399048036,239788778760,937536139764,3669179504364,14373144873774
%N A092765 Consider the 1-D random walk with jumps to next-nearest neighbors. Sequence 
               gives number of paths of length n ending at origin.
%C A092765 In Lakatos-Lindenberg and Shuler besides some physical background there 
               is an exact algebraic expression for the generating function.
%C A092765 Examples from Banderier and Flajolet deal with constrained walks ("meanders" 
               and "excursions") while this sequence counts unrestricted paths.
%D A092765 C. Banderier and P. Flajolet, Basic analytic combinatorics of directed 
               lattice paths. Theoretical Computer Science, Vol. 281:1-2, pp. 37-80, 
               2002
%D A092765 K. Lakatos-Lindenberg and K. E. Shuler, Random walks with nonnearest 
               neighbor transitions. I. Analytic 1-D theory for next-nearest neighbor 
               and exponentially distributed steps, Journal of Mathematical Physics, 
               Vol. 12 Num.4, pp. 633-652, 1971.
%H A092765 P. Flajolet, <a href="http://algo.inria.fr/flajolet/Publications/">Basic 
               analytic combinatorics of directed lattice paths</a>.
%H A092765 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               EightCurve.html">Eight Curve</a>
%F A092765 G.f.: in Maple notation; {x*(1+6*x)*(1-4*x)*(4+9*x)*diff(G(x), x, x)=2*(270*x^3+84*x^2+13*x-1)*diff(G(x), 
               x)+4*x*(12+27*x)*G(x), G(0)=1, D(G)(0)=0} rec; 2*(n+1)*(2*n+1)*a(n+1)+n*(17*n-43)*a(n)=(78*n^2-66*n+36)*a\
               (n-1)+(216*n^2-540*n+324)*a(n-2)
%F A092765 GFun gives the following algebraic equation for generating function: 
               x+2*(1-4*x)*(3*x-2)*g(x)^2+(1-4*x)^2*(9*x+4)*g(x)^4=0 - Sergey Perepechko 
               (persn(AT)aport.ru), Sep 06 2004
%F A092765 a(n) = (2^(2n+1) / Pi) * Int(cos(t)^n*cos(3*t)^n, t=0..Pi/2); a(n) = 
               Sum(binomial(n,k)*binomial(4*n-2*k,2*n-k)*(-3)^k, k=0..n). G.f.: 
               (1 + sqrt(1-4*x)) / ( sqrt(1-4*x) * ( sqrt(1+6*x+2*sqrt(9*x^2+4*x)) 
               + sqrt(1+6*x-2*sqrt(9*x^2+4*x)) ) ). - Max Alekseyev (maxale(AT)gmail.com), 
               Apr 19 2006
%F A092765 a(n) = Sum(binomial(n,k)*binomial(n,2*n-3*k), k=0..n). - Max Alekseyev, 
               Feb 08 2008
%F A092765 a(n) = Sum_{k=0..2n} (-1)^k*C(2n,k)*A027907(n,k) where A027907 is the 
               triangle of trinomial coefficients. [From Paul D. Hanna (pauldhanna(AT)juno.com), 
               Nov 30 2009]
%e A092765 a(3)=6 because 0=+2-1-1, 0=-2+1+1, 0=-1-1+2, 0=+1+1-2, 0=+1-2+1, 0=-1+2-1.
%p A092765 a:=array(0..20):a[0]:=1:a[1]:=0:a[2]:=4:for n from 2 to 19 do a[n+1]:=(-n*(17*n-43)*a[n]+(78*n^2-66*n+36)*a[n\
               -1]+(216*n^2-540*n+324)*a[n-2])/(2*(n+1)*(2*n+1)):print(n+1,a[n+1]) 
               od:
%o A092765 (PARI) a(n) = sum(k=0,n,binomial(n,k)*binomial(4*n-2*k,2*n-k)*(-3)^k) 
               - Max Alekseyev (maxale(AT)gmail.com), Apr 19 2006
%o A092765 (PARI) { a(n)=sum(k=0,n,binomial(n,k)*binomial(n,2*n-3*k)) } - Max Alekseyev, 
               Feb 08 2008
%o A092765 (PARI) {a(n)=sum(k=0,2*n,(-1)^k*binomial(2*n,k)*polcoeff((1+x+x^2)^n,
               k))} [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 30 2009]
%Y A092765 Cf. A027907. [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 30 2009]
%Y A092765 Sequence in context: A137021 A175061 A092187 this_sequence A056315 A103234 
               A074061
%Y A092765 Adjacent sequences: A092762 A092763 A092764 this_sequence A092766 A092767 
               A092768
%K A092765 nonn,new
%O A092765 0,3
%A A092765 Sergey Perepechko (persn(AT)aport.ru), Apr 19 2004
%E A092765 More terms from Max Alekseyev (maxale(AT)gmail.com), Apr 19 2006