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%I #26 Mar 22 2022 09:59:46
%S 2,48,1200,31920,890820,25768512,766053288,23265871200,718834982580,
%T 22523567008800,714044153702880,22861678250567520,738191825153055000,
%U 24011251877148076800,786038700362427057600,25877760367136497398720
%N Number of walks of length 2n+2 from origin to (1,1,0) on a cubic lattice.
%C a(n) is the number of walks of length 2n+2 on a cubic lattice that begin at the origin and end at (1,1,0) using steps (1,0,0), (-1,0,0), (0,1,0), (0,-1,0), (0,0,1), (0,0,-1).
%H G. C. Greubel, <a href="/A135394/b135394.txt">Table of n, a(n) for n = 0..100</a>
%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(2n+2,n) * Sum_{k=0..n} ( binomial(n,k) * binomial(n+2,k+1) * binomial(2k+1,k) ).
%F 6*(n+2)^2*a(n) = (n+1)*((7*n+11)*A002896(n+1)-18*(2*n+1)*A002896(n)). - _Sergey Perepechko_, Feb 08 2011
%F G.f.: (1/(12*x))*(-2*(4*x-1)*(36*x-1)*x*g'' + (-720*x^2+160*x-3)*g' + (-144*x+18)*g) where g is the o.g.f. of A002896. - _Mark van Hoeij_, Nov 12 2011
%F a(n) ~ 3^(7/2) * 36^n / (Pi^(3/2) * n^(3/2)). - _Vaclav Kotesovec_, Nov 27 2017
%t Table[Binomial[2*n + 2, n]*Sum[Binomial[n, k]*Binomial[n + 2, k + 1]*Binomial[2*k + 1, k], {k, 0, n}], {n, 0, 25}] (* _G. C. Greubel_, Oct 12 2016 *)
%o (Maxima) a(n) = binomial(2n+2,n) * sum( binomial(n,k) * binomial(n+2,k+1) * binomial(2k+1,k), k, 0, n )
%o (PARI) a(n) = binomial(2*n+2,n) * sum(k=0,n, binomial(n,k) * binomial(n+2,k+1) * binomial(2*k+1,k)) \\ _Charles R Greathouse IV_, Oct 12 2016
%Y Cf. A002896.
%K easy,nonn,walk
%O 0,1
%A Stefan Hollos (stefan(AT)exstrom.com), Dec 11 2007
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