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A230832 Number of permutations of [n] with exactly one occurrence of the consecutive step pattern up, down, up, down.

%I A230832 #18 Dec 22 2020 03:54:19
%S A230832 0,0,0,0,0,16,192,1472,12800,132352,1366016,14781952,178102272,
%T A230832 2282645504,30639611904,440041603072,6720063012864,107722700685312,
%U A230832 1818098902499328,32319047553515520,601556224722337792,11702621573275975680,237913839294912397312
%N A230832 Number of permutations of [n] with exactly one occurrence of the consecutive step pattern up, down, up, down.
%H A230832 Alois P. Heinz, <a href="/A230832/b230832.txt">Table of n, a(n) for n = 0..460</a>
%F A230832 a(n) ~ c * d^n * n! * n, where d = 0.87361286073825385348141673848..., c = 0.2252746... . - _Vaclav Kotesovec_, Aug 28 2014
%e A230832 a(5) = 16: 13254, 14253, 14352, 15243, 15342, 23154, 24153, 24351, 25143, 25341, 34152, 34251, 35142, 35241, 45132, 45231.
%e A230832 a(6) = 192: 124365, 125364, 125463, ..., 635241, 645132, 645231.
%e A230832 a(7) = 1472: 1235476, 1236475, 1236574, ..., 7635241, 7645132, 7645231.
%p A230832 b:= proc(u, o, t) option remember; `if`(t=9, 0,
%p A230832       `if`(u+o=0, `if`(t>4, 1, 0),
%p A230832       add(b(u-j, o+j-1, [1, 3, 1, 5, 7, 9, 7, 5][t]), j=1..u)+
%p A230832       add(b(u+j-1, o-j, [2, 2, 4, 2, 6, 8, 8, 8][t]), j=1..o)))
%p A230832     end:
%p A230832 a:= n-> b(n, 0, 1):
%p A230832 seq(a(n), n=0..25);
%t A230832 b[u_, o_, t_] := b[u, o, t] = If[t == 9, 0,
%t A230832     If[u + o == 0, If[t > 4, 1,   0],
%t A230832     Sum[b[u - j, o + j - 1, {1, 3, 1, 5, 7, 9, 7, 5}[[t]]], {j, 1, u}] +
%t A230832     Sum[b[u + j - 1, o - j, {2, 2, 4, 2, 6, 8, 8, 8}[[t]]], {j, 1, o}]]];
%t A230832 a[n_] := b[n, 0, 1];
%t A230832 a /@ Range[0, 25] (* after _Alois P. Heinz_ *)
%Y A230832 Column k=1 of A230797.
%K A230832 nonn
%O A230832 0,6
%A A230832 _Alois P. Heinz_, Oct 30 2013