cp's OEIS Frontend

A110711 Number of linear arrangements of n blue, n red and n green items such that first and last elements have the same color but there are no adjacent items of the same color.

%I A110711 #20 Jan 12 2019 20:44:08
%S A110711 0,6,42,288,1992,13980,99432,715344,5196336,38056284,280658100,
%T A110711 2082218160,15528409920,116331315360,874985339760,6604555554720,
%U A110711 50010373864416,379760762209692,2891169309592548,22062102167330592
%N A110711 Number of linear arrangements of n blue, n red and n green items such that first and last elements have the same color but there are no adjacent items of the same color.
%C A110711 The number of linear arrangements is given by A110706 (first and last elements are not adjacent) and A110707 (first and last elements are adjacent) and the number of circular arrangements (counted up to rotations) is given by A110710.
%F A110711 a(n) = 6 * Sum_{k=0..floor(n/2)} binomial(n-1, k) * ( binomial(n-1, k)*binomial(2n-1-2k, n+1) + binomial(n-1, k+1)*binomial(2n-2k-2, n+1) ).
%F A110711 a(n) = A110706(n) - A110707(n).
%F A110711 a(n) = ((n-3)*A000172(n-1) + n*A000172(n))/(n+1). - _Mark van Hoeij_, Jul 14 2010
%F A110711 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_, Nov 01 2015
%p A110711 ogf := 6*((x-2)*hypergeom([1/3,1/3],[1], 27*x^2/((8*x-1)*(x+1)^2)) + 2*hypergeom([1/3,1/3],[2], 27*x^2/((8*x-1)*(x+1)^2))) / ((1-2* x)*(1+x)^(2/3)*(1-8*x)^(1/3));
%p A110711 series(ogf, x=0, 30); # _Mark van Hoeij_, Jan 22 2013
%o A110711 (PARI) a(n) = 6 * sum(k=0,n\2, binomial(n-1,k) * ( binomial(n-1,k)*binomial(2*n-1-2*k,n+1) + binomial(n-1,k+1)*binomial(2*n-2*k-2,n+1) ))
%Y A110711 Cf. A110706, A110707, A110710.
%K A110711 nonn
%O A110711 1,2
%A A110711 _Max Alekseyev_, Aug 04 2005