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%I #18 Oct 08 2018 11:01:42
%S 1,11,181,3499,73501,1623467,37081045,867484331,20661914989,
%T 499049420011,12188943245909,300438089843371,7461880085538581,
%U 186524863637339819,4688354828111460181,118407620161890380459,3002994055439841324301,76441823131542496027499,1952230701520399696996501,50003999526279431605603499
%N a(n) = Sum_{k=1..n} ((3*n-k-1)/(2*n-k))*(3*n-k-2)!/((n-1)!*(n-1)!*(n-k)!).
%C Number of atomic permutations with three runs of equal length n.
%H C. J. Fewster, D. Siemssen, <a href="http://arxiv.org/abs/1403.1723">Enumerating Permutations by their Run Structure</a>, arXiv preprint arXiv:1403.1723 [math.CO], 2014.
%F Conjecture: -(2*n-1)*(n-1)^2*a(n) +2*(32*n^3-131*n^2+187*n-94)*a(n-1) +3*(-86*n^3+721*n^2-1896*n+1617)*a(n-2) -18*(2*n-5)*(3*n-8)*(3*n-7)*a(n-3)=0. - _R. J. Mathar_, Aug 26 2014
%p A241193:=n->add( ((3*n-k-1)/(2*n-k))*(3*n-k-2)!/((n-1)!*(n-1)!*(n-k)!), k=1..n);
%p [seq(A241193(n),n=1..40)];
%t a[n_] := Sum[((3n-k-1)/(2n-k))(3n-k-2)!/((n-1)! (n-1)! (n-k)!), {k, 1, n}];
%t Array[a, 20] (* _Jean-François Alcover_, Oct 08 2018 *)
%o (PARI) a(n) = sum(k=1, n, ((3*n-k-1)/(2*n-k))*(3*n-k-2)!/((n-1)!*(n-1)!*(n-k)!));
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Apr 26 2014
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