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%I #16 Aug 21 2020 17:33:57
%S 1,1,2,4,11,42,210,1292,9352,77505,722294,7470003,84854788,1049924370,
%T 14052654158,202271440732,3115338658280,51118336314648,
%U 890201500701303,16397264064993185,318505677099378561,6506565509515408206,139449260758011488550,3128599281190613701180
%N Number of inversion sequences of length n avoiding the consecutive pattern 001.
%C A length n inversion sequence e_1e_2...e_n is a sequence of integers such that 0 <= e_i < i. The term a(n) counts the inversion sequences of length n with no entries e_i, e_{i+1}, e_{i+2} such that e_i = e_{i+1} < e_{i+2}. That is, a(n) counts the inversion sequences of length n avoiding the consecutive pattern 001.
%H Vaclav Kotesovec, <a href="/A328437/b328437.txt">Table of n, a(n) for n = 0..448</a>
%H Juan S. Auli, <a href="https://search.proquest.com/openview/3f0cef1fbdb016d61e16412e4b855969/1">Pattern Avoidance in Inversion Sequences</a>, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
%H Juan S. Auli and Sergi Elizalde, <a href="https://arxiv.org/abs/1904.02694">Consecutive Patterns in Inversion Sequences</a>, arXiv:1904.02694 [math.CO], 2019.
%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 / sqrt(n), where c = 0.549342310436989831962783548104445992522... - _Vaclav Kotesovec_, Oct 18 2019
%e The a(4)=11 length 4 inversion sequences avoiding the consecutive pattern 001 are 0000, 0100, 0110, 0120, 0101, 0111, 0121, 0102, 0122, 0103, and 0123.
%p # after _Alois P. Heinz_ in A328357
%p b := proc(n, x, t) option remember; `if`(n = 0, 1, add(
%p `if`(t and i = x, 0, b(n - 1, i, i < x)), i = 0 .. n - 1))
%p end proc:
%p a := n -> b(n, -1, false):
%p seq(a(n), n = 0 .. 24);
%t b[n_, x_, t_] := b[n, x, t] = If[n == 0, 1, Sum[If[t && i == x, 0, b[n - 1, i, i < x]], {i, 0, n - 1}]];
%t a[n_] := b[n, -1, False];
%t a /@ Range[0, 24] (* _Jean-François Alcover_, Mar 02 2020, after _Alois P. Heinz_ in A328357 *)
%Y Cf. A328357, A328358, A328429, A328430, A328431, A328432, A328433, A328434, A328435, A328436, A328438, A328439, A328440, A328441, A328442.
%K nonn
%O 0,3
%A _Juan S. Auli_, Oct 17 2019
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