A246246 Number of permutations of [n] with exactly one occurrence of the consecutive step pattern up, down, down.

3, 30, 270, 2322, 20772, 195372, 1958337, 20933154, 238789782, 2900868876, 37451986200, 512534035080, 7416327050415, 113185393797510, 1817654015037150, 30647027466113094, 541407973316966604, 10001886705503187732, 192877025408450517501, 3876090406516703418282

Offset: 4

Author

Alois P. Heinz, Aug 20 2014

Keywords

• nonn

Links

• Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 4..300 (first 160 terms from Alois P. Heinz)

Crossrefs

Column k=1 of A242819.

Programs

Maple

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
      add(b(u-j, o+j-1, [1, 3, 1][t])*`if`(t=3, x, 1), j=1..u)+
      add(b(u+j-1, o-j, 2), j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
seq([T(n)][2], n=4..20); # Vaclav Kotesovec, Aug 22 2014 after Alois P. Heinz

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Expand[Sum[b[u - j, o + j - 1, {1, 3, 1}[[t]]]*If[t == 3, x, 1], {j, 1, u}] + Sum[b[u + j - 1, o - j, 2], {j, 1, o}]]];
a[n_] := Coefficient[b[n, 0, 1], x, 1];
a /@ Range[4, 20] (* Jean-François Alcover, Dec 28 2020, after Maple *)

Formula

a(n) ~ c * (3*sqrt(3)/(2*Pi))^n * n! * n, where c = 0.6335500498606750386938465... = c0 * (c0-1)/3, and c0 = (1+exp(Pi/sqrt(3))) * sqrt(3) / (2*Pi). - Vaclav Kotesovec, Aug 22 2014