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%I #14 Jan 11 2021 23:17:17
%S 0,0,0,1,4,10,28,61,152,318,748,1538,3496,7124,15832,32093,70192,
%T 141814,306508,617878,1323272,2663340,5662600,11383986,24061264,
%U 48330540,101653368,204049636,427414672,857503784,1789891888,3589478621,7469802592,14974962854,31081371148
%N Total number of consecutive triples matching the pattern 132 in all faro permutations of length n.
%C 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. Also the popularity of consecutive pattern 213.
%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.
%F G.f.: x*(-1+4*x^2+2*x+(1-2*x)*sqrt(1-4*x^2)) / ((1-2*x)*(1+sqrt(1-4*x^2))*sqrt(1-4*x^2)).
%e For n = 4, there are 6 faro permutations: 1234, 1243, 1324, 2134, 2143, 3142. They contain in total 4 consecutive patterns 132 and also 4 consecutive patterns 213.
%o (PARI) seq(n)={my(t=sqrt(1-4*x^2+O(x^n))); Vec(x*(-1+4*x^2+2*x+(1-2*x)*t) / ((1-2*x)*(1+t)*t), -(1+n))} \\ _Andrew Howroyd_, Jan 11 2021
%Y A001405 counts faro permutations of length n.
%Y Cf. A107373, A340567, A340569.
%K nonn
%O 0,5
%A _Sergey Kirgizov_, Jan 11 2021