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a(n) = Sum_{k=0..n-2} A205497(n, k) * (k mod 2).
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%I #9 Jun 17 2024 07:56:30

%S 0,0,0,1,3,8,28,136,715,3968,24928,176896,1358611,11184128,99463620,

%T 951878656,9704336283,104932671488,1202007133768,14544442556416,

%U 185212683647587,2475749026562048,34672375957634412,507711943253426176,7757454418668014443,123460740095103991808

%N a(n) = Sum_{k=0..n-2} A205497(n, k) * (k mod 2).

%C Number of linear extensions in L(eps Z_n) that have an odd number of descents. (See Petersen and Yan Zhuang, p. 6.)

%H T. Kyle Petersen and Yan Zhuang, <a href="https://arxiv.org/abs/2403.07181">Zig-zag Eulerian polynomials</a>, arXiv:2403.07181 [math.CO], 2024.

%F a + A373752 = A000111.

%p enum := L -> ListTools:-Enumerate(L):

%p seq(add(c[2]*(1-irem(c[1], 2)), c = enum([A205497row(n)])), n = 0..25);

%Y Cf. A205497, A000111, A373752.

%K nonn

%O 0,5

%A _Peter Luschny_, Jun 16 2024