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%I #23 Jul 30 2023 09:04:35
%S 1,1,2,4,10,26,80,272,1076,4848,24832,142340,902440,6257264,47042924,
%T 380800060,3300070116,30472353884,298595442140
%N Number of permutations of {1,2,3,...,n} each with the same up-down signature as its inverse permutation.
%C An "up-down signature" of the permutation (p(1),p(2),p(3),...,p(n)) is the sequence of (n-1) signs (+ or -) where the k-th sign in the sequence is the sign of (p(k+1)-p(k)). All permutations of {1,2,3,...,n} which are their own inverses obviously have the same up-down signature as their inverse permutations.
%C The terms were calculated by _R. J. Mathar_ (see link).
%H R. J. Mathar, <a href="/A135410/a135410.txt">C++ program</a>
%e The up-down signature of the permutation (4,2,6,3,5,1) is -+-+-. This permutation's inverse permutation is (6,2,4,1,5,3). The inverse permutation also has the up-down signature -+-+-. So these two permutations, among others, are included in the count for n = 6.
%Y Cf. A000085.
%K nonn,more
%O 0,3
%A _Leroy Quet_, Dec 10 2007
%E a(13)-a(14) from _Alois P. Heinz_, Oct 18 2013
%E a(15)-a(18) from _Martin Ehrenstein_, Jul 30 2023
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