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%I #61 Sep 09 2023 18:30:06
%S 1,6,30,174,1092,7188,48852,339720,2403588,17236524,124948668,
%T 913820460,6732898800,49918950240,372104853600,2786716100592,
%U 20955408717396,158149624268220,1197390368733804,9091866006950892,69214297980023256,528150412279712856
%N Number of linear arrangements of n blue, n red and n green items such that there are no adjacent items of the same color.
%C The number of circular arrangements is given by A110707 and A110710.
%H Seiichi Manyama, <a href="/A110706/b110706.txt">Table of n, a(n) for n = 0..1110</a> (terms 1..200 from Vincenzo Librandi)
%H Max Alekseyev, <a href="http://home.gwu.edu/~maxal/gpscripts/">PARI/GP Scripts for Miscellaneous Math Problems</a>.
%H L. Q. Eifler, K. B. Reid Jr., D. P. Roselle, <a href="https://dx.doi.org/10.1007/BF01819761">Sequences with adjacent elements unequal</a>, Aequationes Mathematicae (1971), 6 (2-3), 256-262.
%F a(n) = 2 *( 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, n+1)) ).
%F a(n) = ((3*n-1)*A000172(n-1) + (3*n+2)*A000172(n))/(n+1).
%F D-finite with recurrence: n*(n+1)*a(n) = (n+1)*(7*n-4)*a(n-1) + 8*(n-2)^2*a(n-2). - _Vaclav Kotesovec_, Oct 18 2012
%F a(n) ~ 9*sqrt(3)*2^(3*n-2)/(Pi*n). - _Vaclav Kotesovec_, Oct 18 2012
%F G.f.: (2-x)*(1-8*x)^(-1/3)*(x+1)^(-2/3)*hypergeom([1/3, 1/3],[1],27*x^2/(8*x-1)/(x+1)^2) + 3*x*(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) - 2. - _Mark van Hoeij_, May 14 2013
%F a(n) = 6*A190917(n) for n >= 1. - _R. J. Mathar_, Nov 01 2015
%p a:= proc(n) option remember; `if`(n<2, 1+5*n,
%p ((7*n-4)*a(n-1)+8*(n-2)^2*a(n-2)/(n+1))/n)
%p end:
%p seq(a(n), n=0..21); # _Alois P. Heinz_, Sep 09 2023
%t Table[2*(Sum[Binomial[n-1,k]*(Binomial[n-1,k]*Binomial[2n+1-2k, n+1]+Binomial[n-1,k+1]*Binomial[2n-2k,n+1]),{k,0,Floor[n/2]}]),{n,1,20}] (* _Vaclav Kotesovec_, Oct 18 2012 *)
%t Table[2 (Binomial[2 n + 1, n + 1] HypergeometricPFQ[{1 - n, 1 - n, 1/2 - n/2, -(n/2)}, {1, -(1/2) - n, -n}, 1] + (n - 1) Binomial[2 n, n + 1] HypergeometricPFQ[{1 - n, 2 - n, 1/2 - n/2, 1 - n/2}, {2, 1/2 - n, -n}, 1]), {n, 10}] (* _Eric W. Weisstein_, May 26 2017 *)
%t RecurrenceTable[{n(n+1)*a[n] == (n+1)*(7*n-4)*a[n-1] +8*(n-2)^2*a[n-2], a[1]==6, a[2]==30}, a, {n, 10}] (* _Eric W. Weisstein_, May 27 2017 *)
%o (PARI) a(n)=2*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,n+1)))
%o (Magma) [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, n+1)): k in [0..Floor(n/2)]]): n in [1..25]]; // _G. C. Greubel_, Nov 24 2018
%o (Sage) [2*sum(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, n+1)) for k in range(1+floor(n/2))) for n in (1..25)] # _G. C. Greubel_, Nov 24 2018
%Y Cf. A110707, A110710.
%K nonn
%O 0,2
%A _Max Alekseyev_, Aug 04 2005
%E a(0)=1 prepended by _Alois P. Heinz_, Sep 09 2023
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