A227883 Number of permutations of [n] with exactly one occurrence of the consecutive step pattern up, down, up.

0, 0, 0, 0, 5, 50, 328, 2154, 16751, 144840, 1314149, 12735722, 134159743, 1519210786, 18272249418, 233231701166, 3159471128588, 45243728569842, 682183513506619, 10807962134238068, 179606706777512992, 3123700853586733882, 56737351453843424893

Offset: 0

Author

Alois P. Heinz, Oct 25 2013

Keywords

• nonn

Examples

			a(4) = 5: 1324, 1423, 2314, 2413, 3412.
a(5) = 50: 12435, 12534, 13245, ..., 52314, 52413, 53412.
		

Links

• Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..460 (first 195 terms from Alois P. Heinz)

Crossrefs

Column k=1 of A227884.

Cf. A177477, A245758.

Programs

Maple

b:= proc(u, o, t) option remember;
      `if`(t=7, 0, `if`(u+o=0, `if`(t in [4, 5, 6], 1, 0),
      add(b(u-j, o+j-1, [1, 3, 1, 5, 6, 6][t]), j=1..u)+
      add(b(u+j-1, o-j, [2, 2, 4, 4, 7, 4][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 == 7, 0, If[u + o == 0, If[4 <= t <= 6, 1, 0],
    Sum[b[u - j, o + j - 1, {1, 3, 1, 5, 6, 6}[[t]]], {j, 1, u}] +
    Sum[b[u + j - 1, o - j, {2, 2, 4, 4, 7, 4}[[t]]], {j, 1, o}]]];
a[n_] := b[n, 0, 1];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n * n! * n, where d = A245758 = 0.782704180171521701844707..., c = 0.575076701401064911213333442496869737011... . - Vaclav Kotesovec, Aug 22 2014