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%I #27 Oct 05 2021 18:34:05
%S 1,2,14,204,5104,195040,10570416,771171296,72871890176,8658173200896,
%T 1263326817241600,222078432102900736,46291130226003357696,
%U 11289508838355990634496,3184719803676823556888576,1028950134725811309304442880,377483869192551997938994315264,156057810922284533544621710639104
%N Number of permutations of length 2*n-1 with no local maxima or minima in even positions.
%H Alois P. Heinz, <a href="/A122647/b122647.txt">Table of n, a(n) for n = 1..246</a>
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H D. E. Knuth, <a href="https://cs.stanford.edu/~knuth/papers/whirlpool.pdf">Whirlpool Permutations</a>, May 05 2020
%H M. La Croix, <a href="https://doi.org/10.1016/j.disc.2006.04.015">A combinatorial proof of a result of Gessel and Greene</a>, Discr. Math., 306 (2006), 2251-2256.
%H Jiaxi Lu and Yuanzhe Ding, <a href="https://arxiv.org/abs/2106.09471">A skeleton model to enumerate standard puzzle sequences</a>, arXiv:2106.09471 [math.CO], 2021.
%F a(n) = A113583(n)/2^(n-1).
%p egf:= (x->1/(1-x*tanh(x))-1)(z/sqrt(2)):
%p a:= n-> (2*n)!/n*coeff(series(egf, z, 2*n+1), z, 2*n):
%p seq(a(n), n=1..18); # _Alois P. Heinz_, Oct 05 2021
%Y CF. A113583, A261683.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Sep 21 2006
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