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%I #31 Oct 05 2021 22:12:33
%S 2,8,84,1632,51040,2340480,147985824,12338740736,1311694023168,
%T 173163464017920,27793189979315200,5329882370469617664,
%U 1203569385876087300096,316106247473967737765888,95541594110304706706657280,32926404311225961897742172160
%N Number of permutations p of {1..2n} such that p[2k-1]<p[2k] <=> p[2k]<p[2k+1].
%C The relation between p[2n-1] and p[2n] is arbitrary; hence a(n) = 2*n*A122647(n). a(n) is also (surprisingly) the number of 2 X n whirlpool permutations (see link, also A334518). - _Don Knuth_, May 06 2020.
%H Alois P. Heinz, <a href="/A261683/b261683.txt">Table of n, a(n) for n = 1..245</a>
%H Nicolas Basset, <a href="https://hal.archives-ouvertes.fr/hal-01093994">Counting and generating permutations in regular classes of permutations</a>, HAL Id: hal-01093994, 2014.
%H D. E. Knuth, <a href="https://cs.stanford.edu/~knuth/papers/whirlpool.pdf">Whirlpool Permutations</a>, May 05 2020.
%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 Basset (2014, Eq. (4)) gives a g.f.
%F a(n) = (2n)! [z^(2n)] 2*sqrt(2)*z*(exp(sqrt(2)*z)-1) / (2+sqrt(2)*z + (2-sqrt(2)*z)*exp(sqrt(2)*z)). - _Alois P. Heinz_, Sep 06 2015
%p egf:= 2*(x->1/(1-x*tanh(x))-1)(z/sqrt(2)):
%p a:= n-> (2*n)!*coeff(series(egf,z,2*n+1),z,2*n):
%p seq(a(n), n=1..20); # _Alois P. Heinz_, Sep 06 2015
%Y Cf. A122647, A334518, A334519.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Sep 05 2015
%E More terms from _Alois P. Heinz_, Sep 06 2015
%E Name corrected by _Don Knuth_. - _N. J. A. Sloane_, May 06 2020
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