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

0, 0, 0, 0, 0, 16, 192, 1472, 12800, 132352, 1366016, 14781952, 178102272, 2282645504, 30639611904, 440041603072, 6720063012864, 107722700685312, 1818098902499328, 32319047553515520, 601556224722337792, 11702621573275975680, 237913839294912397312

Offset: 0

Author

Alois P. Heinz, Oct 30 2013

Keywords

• nonn

Examples

			a(5) = 16: 13254, 14253, 14352, 15243, 15342, 23154, 24153, 24351, 25143, 25341, 34152, 34251, 35142, 35241, 45132, 45231.
a(6) = 192: 124365, 125364, 125463, ..., 635241, 645132, 645231.
a(7) = 1472: 1235476, 1236475, 1236574, ..., 7635241, 7645132, 7645231.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..460

Crossrefs

Column k=1 of A230797.

Programs

Maple

b:= proc(u, o, t) option remember; `if`(t=9, 0,
      `if`(u+o=0, `if`(t>4, 1, 0),
      add(b(u-j, o+j-1, [1, 3, 1, 5, 7, 9, 7, 5][t]), j=1..u)+
      add(b(u+j-1, o-j, [2, 2, 4, 2, 6, 8, 8, 8][t]), j=1..o)))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[t == 9, 0,
    If[u + o == 0, If[t > 4, 1,   0],
    Sum[b[u - j, o + j - 1, {1, 3, 1, 5, 7, 9, 7, 5}[[t]]], {j, 1, u}] +
    Sum[b[u + j - 1, o - j, {2, 2, 4, 2, 6, 8, 8, 8}[[t]]], {j, 1, o}]]];
a[n_] := b[n, 0, 1];
a /@ Range[0, 25] (* after Alois P. Heinz *)

Formula

a(n) ~ c * d^n * n! * n, where d = 0.87361286073825385348141673848..., c = 0.2252746... . - Vaclav Kotesovec, Aug 28 2014