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A177471 Avoiding the pattern 121'2'. To avoid 121'2' means not to have four consecutive letters such that the first letter is less than the second one and the third letter is less than the fourth one.

%I A177471 #18 Nov 03 2020 04:57:43
%S A177471 1,1,2,6,18,61,281,1541,8920,57924,437490,3611389,31721537,304085783,
%T A177471 3180772870,35422074330,418050879810,5266547286061,70459362412265,
%U A177471 991921937012273,14681437097585260,228615478225446360,3730868960721027906,63577641238069645741
%N A177471 Avoiding the pattern 121'2'. To avoid 121'2' means not to have four consecutive letters such that the first letter is less than the second one and the third letter is less than the fourth one.
%H A177471 Alois P. Heinz, <a href="/A177471/b177471.txt">Table of n, a(n) for n = 0..472</a>
%H A177471 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 A177471 a(n) ~ c * d^n * n!, where d = 0.7411900298994603810165..., c = 2.41202786457703060749584... . - _Vaclav Kotesovec_, Aug 22 2014
%p A177471 b:= proc(u, o, s, t) option remember; `if`(u+o=0, 1,
%p A177471        add(b(u-j, o+j-1, t, false), j=1..u)+
%p A177471       `if`(s, 0, add(b(u+j-1, o-j, t, true), j=1..o)))
%p A177471     end:
%p A177471 a:= n-> b(n, 0, false$2):
%p A177471 seq(a(n), n=0..25);  # _Alois P. Heinz_, Oct 25 2013
%t A177471 b[u_, o_, s_, t_] := b[u, o, s, t] = If[u+o == 0, 1, Sum[b[u-j, o+j-1, t, False], {j, 1, u}] + If[s, 0, Sum[b[u+j-1, o-j, t, True], {j, 1, o}]]];
%t A177471 a[n_] := b[n, 0, False, False];
%t A177471 a /@ Range[0, 25] (* _Jean-François Alcover_, Nov 03 2020, after _Alois P. Heinz_ *)
%K A177471 nonn
%O A177471 0,3
%A A177471 Signy Olafsdottir (signy06(AT)ru.is), May 09 2010
%E A177471 a(10)-a(23) from _Alois P. Heinz_, Oct 25 2013