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%I #19 Mar 01 2020 04:23:39
%S 1,1,2,5,17,69,330,1797,11028,74932,559351,4540088,39840318,375421225,
%T 3782383945,40548234374,460956742449,5536790753853,70077462043662,
%U 931945968071778,12993337101354500,189485727877247991,2884989393948284323,45772604755492432599
%N Number of inversion sequences of length n where all consecutive subsequences i,j,k satisfy i >= j <= k or i <= j >= k.
%H Alois P. Heinz, <a href="/A326412/b326412.txt">Table of n, a(n) for n = 0..483</a>
%H 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 a(n) ~ n! * c * 2^n * n^((Pi+1)/2) / Pi^n, where c = 0.0662002484840446134... - _Vaclav Kotesovec_, Oct 31 2019
%e a(4) = 17: 0000, 0001, 0002, 0003, 0010, 0011, 0020, 0021, 0022, 0100, 0101, 0102, 0103, 0110, 0111, 0112, 0113.
%p b:= proc(n, j, t, u, c) option remember; `if`(n=0, 1, add(
%p `if`(c>0 or i>=j and t or i<=j and u, b(n-1, i,
%p is(i<=j), is(i>=j), max(0, c-1)), 0), i=1..n))
%p end:
%p a:= n-> b(n, 0, true$2, 2):
%p seq(a(n), n=0..25);
%t 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 a[n_] := b[n, 0, True, True, 2];
%t a /@ Range[0, 25] (* _Jean-François Alcover_, Mar 01 2020, after _Alois P. Heinz_ *)
%Y Cf. A000108, A000142, A000225, A001250, A328357, A328358, A328409, A328425, A328491, A326308.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Oct 17 2019
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