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A230621 Number of permutations of [n] with exactly two (possibly overlapping) occurrences of the consecutive step pattern {up}^2. 3
0, 0, 0, 0, 1, 8, 86, 803, 8221, 86214, 966114, 11405511, 142934124, 1892755874, 26487024478, 390658292572, 6063383527327, 98824236282650, 1688354110698402, 30179347977813309, 563462569163994435, 10970288500929001986, 222384832378410907480 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..462

FORMULA

a(n) ~ c * (3*sqrt(3)/(2*Pi))^n * n! * n^2, where c = 0.0359701024355206... . - Vaclav Kotesovec, Sep 06 2014

EXAMPLE

a(4) = 1: 1234.

a(5) = 8: 12354, 12453, 13452, 21345, 23451, 31245, 41235, 51234.

a(6) = 86: 123546, 123645, 123654, ..., 631245, 641235, 651234.

a(7) = 803: 1235476, 1236475, 1236547, ..., 7631245, 7641235, 7651234.

MAPLE

b:= proc(u, o, t) option remember;

      `if`(t=7, 0, `if`(u+o=0, `if`(t in [4, 6], 1, 0),

      add(b(u-j, o+j-1, [1, 1, 5, 6, 5, 6][t]), j=1..u)+

      add(b(u+j-1, o-j, [2, 3, 4, 7, 3, 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[u + o == 0, 1,

    Sum[b[u - j, o + j - 1, 1], {j, 1, u}] +

    Sum[b[u + j - 1, o - j, 2]*If[t == 2, x, 1], {j, 1, o}] // Expand];

a[n_] := Coefficient[b[n, 0, 1], x, 2];

a /@ Range[0, 25] (* Jean-Fran├žois Alcover, Dec 21 2020, after Alois P. Heinz in A162975 *)

CROSSREFS

Column k=2 of A162975.

Cf. A230620.

Sequence in context: A349334 A261501 A180582 * A268052 A268075 A202545

Adjacent sequences:  A230618 A230619 A230620 * A230622 A230623 A230624

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Oct 25 2013

STATUS

approved

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Last modified May 23 07:54 EDT 2022. Contains 353961 sequences. (Running on oeis4.)