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A110711
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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.
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5
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0, 6, 42, 288, 1992, 13980, 99432, 715344, 5196336, 38056284, 280658100, 2082218160, 15528409920, 116331315360, 874985339760, 6604555554720, 50010373864416, 379760762209692, 2891169309592548, 22062102167330592
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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) ).
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
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MAPLE
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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));
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PROG
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(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) ))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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