%I #34 Apr 14 2023 14:28:15
%S 1,1,2,6,21,77,287,1079,4082,15522,59280,227240,873886,3370030,
%T 13027730,50469890,195892565,761615285,2965576715,11563073315,
%U 45141073925,176423482325,690215089745,2702831489825,10593202603775,41550902139551,163099562175851
%N Number of length n inversion sequences avoiding the patterns 101, 102, 201, and 210.
%C A length n inversion sequence e_1e_2...e_n is a sequence of integers where 0 <= e_i <= i-1. The term a(n) counts those length n inversion sequences with no entries e_i, e_j, e_k (where i<j<k) such that e_i > e_j <> e_k. This is the same as the set of length n inversion sequences avoiding 101, 102, 201, and 210.
%C It is conjectured that a_n also counts those length n inversion sequences with no entries e_i, e_j, e_k (where i<j<k) such that e_i < e_j > e_k and e_i <> e_k. This is the same as the set of length n inversion sequences avoiding 021 and 120.
%H Alois P. Heinz, <a href="/A279561/b279561.txt">Table of n, a(n) for n = 0..1664</a>
%H Shane Chern, <a href="https://arxiv.org/abs/2006.04318">On 0012-avoiding inversion sequences and a Conjecture of Lin and Ma</a>, arXiv:2006.04318 [math.CO], 2020.
%H A. V. Kitaev and A. Vartanian, <a href="https://arxiv.org/abs/2304.05671">Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter</a>, arXiv:2304.05671 [math.CA], 2023.
%H Megan A. Martinez and Carla D. Savage, <a href="https://arxiv.org/abs/1609.08106">Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations</a>, arXiv:1609.08106 [math.CO], 2016.
%H Chunyan Yan and Zhicong Lin, <a href="https://arxiv.org/abs/1912.03674">Inversion sequences avoiding pairs of patterns</a>, arXiv:1912.03674 [math.CO], 2019.
%F a(n) = 1 + Sum_{i=1..n-1} binomial(2i, i-1).
%F a(n) = 1 + A057552(n-2).
%F G.f.: (1-4*x+sqrt(-16*x^3+20*x^2-8*x+1))/(2*(x-1)*(4*x-1)).
%F D-finite with recurrence: n*a(n) +(-7*n+6)*a(n-1) +2*(7*n-13)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - _R. J. Mathar_, Feb 21 2020
%e The length 4 inversion sequences avoiding (101, 102, 201, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0100, 0110, 0111, 0112, 0113, 0120, 0121, 0122, 0123.
%e The length 4 inversion sequences avoiding (021, 120) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0022, 0023, 0100, 0101, 0102, 0103, 0110, 0111, 0112, 0113, 0122, 0123.
%p a:= proc(n) option remember; `if`(n<3, 1+n*(n-1)/2,
%p ((5*n^2-12*n+6)*a(n-1)-(4*n^2-10*n+6)*a(n-2))/((n-2)*n))
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jan 18 2017
%t a[n_] := 1 + Sum[Binomial[2i, i-1], {i, 0, n-1}];
%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 28 2017 *)
%Y Cf. A000108, A057552, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.
%K nonn
%O 0,3
%A _Megan A. Martinez_, Jan 17 2017