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%I #31 Mar 08 2022 10:21:33
%S 1,1,2,6,19,70,331,1863,11637,81110,635550,5495339,51590494,524043395,
%T 5743546943,67478821537,844983073638,11240221721390,158365579448315,
%U 2355375055596386,36870671943986643,606008531691619131,10435226671431973345,187860338952519968538
%N Number of permutations of 1..n avoiding adjacent step pattern up, down, up.
%C Suppose a < b, c < b, and c < d. To avoid abcd means not to have four consecutive letters such that the first letter is less than the second one, the third letter is less than the second one, and the third letter is less than the last one.
%H Alois P. Heinz, <a href="/A177477/b177477.txt">Table of n, a(n) for n = 0..468</a>
%H Sergey 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 a(n) ~ c * d^n * n!, where d = A245758 = 0.7827041801715217018447074977..., c = 2.035127405829990832658061124449458067... . - _Vaclav Kotesovec_, Aug 22 2014
%p b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p add(b(u-j, o+j-1, [1, 3, 1][t]), j=1..u)+
%p `if`(t=3, 0, add(b(u+j-1, o-j, 2), j=1..o)))
%p end:
%p a:= n-> b(n, 0, 1):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Mar 10 2020
%t b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
%t Sum[b[u - j, o + j - 1, {1, 3, 1}[[t]]], {j, 1, u}] +
%t If[t == 3, 0, Sum[b[u + j - 1, o - j, 2], {j, 1, o}]]];
%t a[n_] := b[n, 0, 1];
%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Mar 08 2022, after _Alois P. Heinz_ *)
%Y Column k=0 of A227884.
%Y Column k=5 of A242784.
%Y Cf. A227883, A245758.
%K nonn
%O 0,3
%A Submitted independently by Signy Olafsdottir (signy06(AT)ru.is), May 09 2010 (9 terms) and _R. H. Hardin_, May 10 2010 (17 terms)
%E a(18)-a(23) from _Alois P. Heinz_, Oct 06 2013
%E a(0)=1 prepended by _Alois P. Heinz_, Mar 10 2020
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