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A332344 a(n) is the number of permutations w of 1,2,...,n such that w is alternating and w^{-1} is reverse alternating.

%I A332344 #8 Feb 11 2020 12:06:51
%S A332344 1,1,0,1,1,3,6,19,56,213,816,3717,17408,92935,513152,3147495,19993728,
%T A332344 138638825,995169664,7700244745,61608152704,526317901451,
%U A332344 4642742078336,43407723925499,418444180284544,4250149857500861,44444888840063360
%N A332344 a(n) is the number of permutations w of 1,2,...,n such that w is alternating and w^{-1} is reverse alternating.
%H A332344 R. P. Stanley, <a href="http://arxiv.org/abs/math/0603520">Alternating permutations and symmetric functions</a>, arXiv:math/0603520 [math.CO], 2006.
%e A332344 The only alternating permutation of 1,2,3,4 whose inverse is reverse alternating is 2413.
%e A332344 The three alternating permutations of 1,...,5 whose inverses are reverse alternating are 24153, 24351 and 45231.
%Y A332344 For odd n, a(n) = A007999(n).
%Y A332344 For even n > 1, a(n) = A007999(n) - A007999(n-2).
%Y A332344 For n > 1, a(n) = A332345(n)/2 - A007999(n).
%K A332344 nonn
%O A332344 0,6
%A A332344 _David Bevan_, Feb 10 2020