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

4,1

Links

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

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

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 *)

Crossrefs

Column k=1 of A242819.

Sequence in context: A120689 A359614 A180426 * A136896 A121085 A144282

Adjacent sequences: A246243 A246244 A246245 * A246247 A246248 A246249

Keyword

nonn

Author

Alois P. Heinz, Aug 20 2014

Status

approved