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%I #15 May 10 2020 06:04:54
%S 0,2,14,96,664,4660,33144,238448,1732112,12685428,93552700,694072720,
%T 5176136640,38777105120,291661779920,2201518518240,16670124621472,
%U 126586920736564,963723103197516,7354034055776864,56236603567496720
%N Number of linear arrangements of n blue, n red and n green items such that first and last elements are blue but there are no adjacent items of the same color.
%H Max Alekseyev, <a href="http://home.gwu.edu/~maxal/gpscripts/">PARI scripts for various problems</a>
%H L. Q. Eifler, K. B. Reid Jr., D. P. Roselle, <a href="http://dx.doi.org/10.1007/BF01819761">Sequences with adjacent elements unequal</a>, Aequationes Mathematicae 6 (2-3), 1971.
%F a(n) = A110711(n) / 3.
%F a(n) = Sum[k=0..[n/2]] binomial(n-1,2k) * binomial(2k,k) * binomial(n-1+k,k+1) * 2^(n-1-2k).
%F G.f.: (2*x-1)^2*(1-8*x)^(-4/3)*(x+1)^(-8/3)*hypergeom([4/3, 4/3],[2],27*x^2/((8*x-1)*(x+1)^2))-(1-8*x)^(-1/3)*(x+1)^(-2/3)*hypergeom([1/3, 1/3],[1],27*x^2/((8*x-1)*(x+1)^2)). - _Mark van Hoeij_, May 14 2013
%F Conjecture: -(n+1)*(n-2)*a(n) +(7*n^2-13*n+4)*a(n-1) +8*(n-2)^2*a(n-2)=0. - _R. J. Mathar_, Jul 23 2014
%o (PARI) { a(n) = sum(k=0,n\2, binomial(n-1,2*k) * binomial(2*k,k) * binomial(n-1+k,k+1) * 2^(n-1-2*k) ) }
%Y Cf. A110706, A110707, A110710, A110711, A141147, A141148.
%K nonn
%O 1,2
%A _Max Alekseyev_, Jun 10 2008