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%I #13 May 13 2023 01:53:18
%S 1,15,310,7455,195426,5416026,156061620,4628393055,140348412490,
%T 4331544836190,135614951248140,4296741195214650,137507314754659500,
%U 4438467396322843500,144329729055650881560,4723733064176346346335
%N Number of walks from origin to (1,0,0) in a cubic lattice.
%C a(n) is the number of walks of length 2n+1 on a cubic lattice that start at the origin and end at (1,0,0) using steps (1,0,0), (-1,0,0), (0,1,0), (0,-1,0), (0,0,1), (0,0,-1).
%H S. Hollos and R. Hollos, <a href="http://www.exstrom.com/math/lattice/latpath.html">Lattice Paths and Walks</a>.
%F a(n) = binomial(2*n+1,n) * Sum_{k=0..n} binomial(n,k) * binomial(n+1,k) * binomial(2*k,k).
%F G.f.: G(z) = 1/6 * (1/sqrt(1+12*z)*hypergeom([1/8,3/8],[1],64/81*z*(1+sqrt(1-36*z))^2*(2+sqrt(1-36*z))^4/(1+12*z)^4)*hypergeom([1/8,3/8],[1],64/81*z*(1-sqrt(1-36*z))^2*(2-sqrt(1-36*z))^4/(1+12*z)^4) - 1). - _Sergey Perepechko_, Jan 31 2011
%F a(n) = A002896(n+1)/6.
%t f[n_] := Binomial[2 n + 1, n]*Sum[ Binomial[n, k]*Binomial[n + 1, k]*Binomial[2 k, k], {k, 0, n}]; Array[f, 16, 0] (* _Robert G. Wilson v_ *)
%o (Maxima) a(n) = binomial(2n+1,n) * sum( binomial(n,k) * binomial(n+1,k) * binomial(2k,k), k, 0, n )
%Y Cf. A002896.
%K easy,nonn
%O 0,2
%A Stefan Hollos (stefan(AT)exstrom.com), Dec 11 2007
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