%I #12 Oct 21 2020 14:17:25
%S 2,2,0,0,11,33,29,60,3905,19524,62879,275436,10187165,71191608,
%T 481719419,3211782240,101121160145,904977244224,10099756468559,
%U 89733565369536,2746252055597525,29900664884062848,479479605967022099,5296351543857279360,166991194742961246905
%N a(n) = 2^floor((2-n)/2)*Sum_{0 <= k <= n and A337966(n, k) > 0} A173018(n, k).
%C The two sequences A337999 and A338000 represent the number of alternating permutations of order n as the difference between the greater and the smaller of the two absolute values (for n >= 2).
%F (a(n) - A338000(n))/2 = (-1)^floor(n/2)*A000111(n).
%F (-1)^floor(n/2)*(a(n) - A338000(n)) = A001250(n) for n >= 2.
%e Row 6 of A337967 is: 1, 57, -302, -302, 57, 1, 0. The sum of positive terms is 2*(1 + 57) = 116. Thus a(6) = 116/4 = 29.
%p A337999 := n -> add(`if`(A337966(n,k)>0,A173018(n,k),0),k=0..n)/2^floor((n-2)/2): seq(A337999(n), n=0..24);
%Y Cf. A337966, A337967, A173018, A338000, A000111, A001250.
%K nonn
%O 0,1
%A _Peter Luschny_, Oct 06 2020