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A177479 Number of permutations of 1..n avoiding adjacent step pattern up, down, down.

%I A177479 #30 Mar 10 2020 11:55:46
%S A177479 1,1,2,6,21,90,450,2619,17334,129114,1067661,9713682,96393726,
%T A177479 1036348587,11998603710,148842430470,1969461102357,27688474234602,
%U A177479 412166988789642,6476330295597051,107117619952992966,1860296912926495938,33845967939906741213,643778989807702357314
%N A177479 Number of permutations of 1..n avoiding adjacent step pattern up, down, down.
%C A177479 Suppose j<i, k<j and k<m. To avoid ijkm means not to have four consecutive letters such that the first letter is larger than the second one, the second letter is larger than the third one, and the fourth letter is larger than the third one.
%H A177479 Alois P. Heinz, <a href="/A177479/b177479.txt">Table of n, a(n) for n = 0..464</a>
%H A177479 S. Kitaev, <a href="https://doi.org/10.1016/j.dam.2006.09.011">Introduction to partially ordered patterns</a>, Discrete Applied Mathematics 155 (2007), 929-944.
%F A177479 E.g.f.: (exp(3*x/2) + 2*cos(sqrt(3)*x/2)) / (3*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)). - _Vaclav Kotesovec_, Aug 23 2014
%F A177479 a(n) ~ n! * (1+exp(Pi/sqrt(3))) * 3^(3*n/2+1/2) / (2*Pi)^(n+1). - _Vaclav Kotesovec_, Aug 23 2014
%p A177479 b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p A177479        add(b(u+j-1, o-j, 1), j=1..o)+ `if`(t<2,
%p A177479        add(b(u-j, o+j-1, `if`(t=0, 0, 2)), j=1..u), 0))
%p A177479     end:
%p A177479 a:= n-> b(n, 0, 0):
%p A177479 seq(a(n), n=0..30);  # _Alois P. Heinz_, Oct 07 2013
%t A177479 FullSimplify[Rest[CoefficientList[Series[(E^(3*x/2) + 2*Cos[Sqrt[3]*x/2]) / (3*Cos[Sqrt[3]*x/2] - Sqrt[3]*Sin[Sqrt[3]*x/2]), {x, 0, 20}], x] * Range[0, 20]!]] (* _Vaclav Kotesovec_, Aug 23 2014 *)
%Y A177479 Column k=0 of A242819.
%Y A177479 Columns k=4,6 of A242784.
%K A177479 nonn
%O A177479 0,3
%A A177479 Submitted independently by Signy Olafsdottir (signy06(AT)ru.is), May 09 2010 (9 terms) and _R. H. Hardin_, May 10 2010 (17 terms)
%E A177479 a(18)-a(23) from _Alois P. Heinz_, Oct 06 2013