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%I #50 Sep 08 2022 08:45:18
%S 0,0,1,2,7,14,38,76,187,374,874,1748,3958,7916,17548,35096,76627,
%T 153254,330818,661636,1415650,2831300,6015316,12030632,25413342,
%U 50826684,106853668,213707336,447472972,894945944,1867450648,3734901296,7770342787,15540685574
%N a(n) = (n/2)*binomial(n-1, floor((n-1)/2)) - 2^(n-2).
%C Total number of descents in all faro permutations of length n-1. Faro permutations are permutations avoiding the three consecutive patterns 231, 321 and 312. They are obtained by a perfect faro shuffle of two nondecreasing words of lengths differing by at most one. See also A340567, A340568 and A340569. - _Sergey Kirgizov_, Jan 11 2021
%H Vincenzo Librandi, <a href="/A107373/b107373.txt">Table of n, a(n) for n = 1..1000</a>
%H Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, <a href="https://arxiv.org/abs/2010.06270">Pattern statistics in faro words and permutations</a>, arXiv:2010.06270 [math.CO], 2020. See Table 1.
%H F. Disanto and S. Rinaldi, <a href="http://www.mat.unisi.it/newsito/puma/public_html/22_1/2-disanto_rinaldi.pdf">Symmetric convex permutominoes and involutions</a>, PU. M. A. 22:1 (2011), 39-60.
%H Igor Pak, <a href="http://www.math.ucla.edu/~pak/papers/ppss3.pdf">The area of cyclic polygons: Recent progress on Robbins' Conjectures</a>, Adv. Applied Math. 34 (2005), 690-696. Special issue in memory of David Robbins.
%F a(2*n) = 2*A000531(n-1); a(2*n+1) = A000531(n). - _Max Alekseyev_, Sep 30 2013
%F (1-n)*a(n) + 2*(n-1)*a(n-1) + 4*(n-2)*a(n-2) + 8*(-n+2)*a(n-3) = 0. - _R. J. Mathar_, May 26 2013
%p A056040 := n -> n!/iquo(n,2)!^2:
%p A133265 := n -> (n+2+(n-2)*(-1)^n)/2:
%p A107373 := n -> (A056040(n)*A133265(n)-2^n)/4:
%p seq(A107373(n),n=1..34); # _Peter Luschny_, Aug 30 2011
%t Table[(n/2) Binomial[n-1, Floor[(n-1)/2]] - 2^(n-2), {n, 1, 40}] (* _Vincenzo Librandi_, Oct 01 2013 *)
%o (Magma) [(n/2)*Binomial(n-1, Floor((n-1)/2)) - 2^(n-2): n in [1..40]]; // _Vincenzo Librandi_, Oct 01 2013
%Y Cf. A131019, A131020, A131021.
%K nonn
%O 1,4
%A _N. J. A. Sloane_, Jul 20 2007
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