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a(n) ~ c * d^n / n^(3/2), where d = 5.98041772076926677236919875200507... is the positive root of the equation -32 - 195*d - 12*d^2 - 112*d^3 + 20*d^4 = 0 and c = 0.1056946795054351807407212356928404107733262398133039312067247126343... - Vaclav Kotesovec, Oct 07 2021
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It was proven shown 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 and e_i > e_k. This is the same as the set of length n inversion sequences avoiding 100, 120, 201, and 210.
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b[n_, i_, l_] := b[n, i, l] = If[n == 0, 1, Sum[b[n-1, i-#+2, j-#+1]& @ Max[1, If[j == l, 0, l]], {j, 1, i}]]; a[n_] := b[n, 1, 1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 10 2017, after Alois P. Heinz *)
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Alois P. Heinz, <a href="/A279568/b279568.txt">Table of n, a(n) for n = 0..1294</a>
b(n-1, i-h+2, j-h+1))(max(1, `if`(j<>l, =l, 0, l))), j=1..i))