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A326308 Number of inversion sequences of length n where all consecutive subsequences i,j,k satisfy i > j < k or i < j > k.

%I A326308 #17 Feb 29 2020 13:45:01
%S A326308 1,1,2,1,3,6,26,85,476,2171,14905,87153,708825,5053464,47514180,
%T A326308 399542814,4264132468,41306091312,493337571005,5408829555639,
%U A326308 71476985762027,874870165668858,12673922434134249,171294209823727623,2699365743596908540,39925463781029750810
%N A326308 Number of inversion sequences of length n where all consecutive subsequences i,j,k satisfy i > j < k or i < j > k.
%H A326308 Alois P. Heinz, <a href="/A326308/b326308.txt">Table of n, a(n) for n = 0..486</a>
%H A326308 Juan S. Auli and Sergi Elizalde, <a href="https://arxiv.org/abs/1906.07365">Consecutive patterns in inversion sequences II: avoiding patterns of relations</a>, arXiv:1906.07365 [math.CO], 2019.
%F A326308 a(n) ~ n! * c * 2^n / (Pi^n * sqrt(n)), where c = 1.0215796642504649172542599982453320786973706265645819484... - _Vaclav Kotesovec_, Oct 31 2019
%e A326308 a(6) = 26: 010101, 010102, 010103, 010104, 010105, 010201, 010202, 010203, 010204, 010205, 010212, 010213, 010214, 010215, 010301, 010302, 010303, 010304, 010305, 010312, 010313, 010314, 010315, 010323, 010324, 010325.
%p A326308 b:= proc(n, j, t, u, c) option remember; `if`(n=0, 1, add(
%p A326308       `if`(c>0 or i>j and t or i<j and u, b(n-1, i,
%p A326308         is(i<j), is(i>j), max(0, c-1)), 0), i=1..n))
%p A326308     end:
%p A326308 a:= n-> b(n, 0, true$2, 2):
%p A326308 seq(a(n), n=0..25);
%t A326308 b[n_, j_, t_, u_, c_] := b[n, j, t, u, c] = If[n == 0, 1, Sum[If[c>0 || i>j && t || i<j && u, b[n-1, i, i<j, i>j, Max[0, c-1]], 0], {i, 1, n}]];
%t A326308 a[n_] := b[n, 0, True, True, 2];
%t A326308 a /@ Range[0, 25] (* _Jean-François Alcover_, Feb 29 2020, after _Alois P. Heinz_ *)
%Y A326308 Cf. A000108, A000142, A000225, A001250, A328357, A328358, A328409, A328425, A328491, A326412.
%K A326308 nonn
%O A326308 0,3
%A A326308 _Alois P. Heinz_, Oct 17 2019