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%I #33 Apr 05 2019 05:42:00
%S 1,2,6,24,104,464,2088,9392,42064,187296,828776,3644912,15937776,
%T 69317984,300009744,1292654304,5547021728,23715100480,101046014952,
%U 429209373296,1817975905456,7680278380512,32368750662320
%N Number of square permutations of length n.
%H Michael Albert, Steve Linton, Nik Ruskuc, Vincent Vatter, Steve Waton, <a href="http://hdl.handle.net/10023/2000">On convex permutations</a>, preprint.
%H Michael Albert, Steve Linton, Nik Ruskuc, Vincent Vatter, Steve Waton, <a href="http://dx.doi.org/10.1016/j.disc.2011.01.009">On convex permutations</a>, Discrete Mathematics, vol.311, pp.715-722, (2011).
%H A. Bernini, F. Disanto, R. Pinzani and S. Rinaldi, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Rinaldi/rinaldi5.html">Permutations defining convex permutominoes</a>, J. Int. Seq. 10 (2007) # 07.9.7.
%H Enrica Duchi, <a href="https://arxiv.org/abs/1904.02691">A code for square permutations and convex permutominoes</a>, arXiv:1904.02691 [math.CO], 2019.
%H Sergey Kitaev and Jeffrey Remmel, <a href="http://arxiv.org/abs/1201.1323">Simple marked mesh patterns</a>, arXiv preprint arXiv:1201.1323 [math.CO], 2012.
%H S. Kitaev, J. Remmel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Kitaev/kitaev5.html">Quadrant Marked Mesh Patterns</a>, J. Int. Seq. 15 (2012) # 12.4.7
%H T. Mansour and S. Severini, <a href="https://arxiv.org/abs/math/0603225">Grid polygons from permutations and their enumeration by the kernel method</a>, arXiv:math/0603225 [math.CO], 2006.
%F a(n) = 2*(n+2) * 4^(n-3) - 4*(2*n-5) * C(2*n-6,n-3) for n>=2, a(1)=1.
%F G.f.: x*(1-6*x+10*x^2-4*x^2*sqrt(1-4*x))/(1-4*x)^2 (See theorem 3.1 in Albert et al. reference). [_Joerg Arndt_, Jun 21 2011]
%F Conjecture: +(n-3)*(n-8)*a(n) +2*(-4*n^2+43*n-96)*a(n-1) +8*(2*n-7)*(n-7)*a(n-2)=0. - _R. J. Mathar_, Oct 16 2017
%t a[1] = 1; a[n_] := 2(n+2) * 4^(n-3) - 4(2n-5) * Binomial[2n-6, n-3];
%t Array[a, 30] (* _Jean-François Alcover_, Jul 22 2018 *)
%o (PARI) a(n) = if(n<=1,n,2*(n+2) * 4^(n-3) - 4*(2*n-5) * binomial(2*n-6,n-3)); /* _Joerg Arndt_, Jun 21 2011 */
%K nonn
%O 1,2
%A _Ralf Stephan_, May 08 2007
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