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.

0, 6, 42, 288, 1992, 13980, 99432, 715344, 5196336, 38056284, 280658100, 2082218160, 15528409920, 116331315360, 874985339760, 6604555554720, 50010373864416, 379760762209692, 2891169309592548, 22062102167330592

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..20.

Formula

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) ).

a(n) = A110706(n) - A110707(n).

a(n) = ((n-3)*A000172(n-1) + n*A000172(n))/(n+1). - Mark van Hoeij, Jul 14 2010

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

Maple

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));
series(ogf, x=0, 30); # Mark van Hoeij, Jan 22 2013

Prog

(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) ))

Crossrefs

Cf. A110706, A110707, A110710.

Sequence in context: A373715 A157335 A057089 * A156361 A216517 A055272

Adjacent sequences: A110708 A110709 A110710 * A110712 A110713 A110714

Keyword

nonn

Author

Max Alekseyev, Aug 04 2005

Status

approved