A193364 Number of permutations that have a fixed point and contain 123.

0, 0, 0, 1, 1, 3, 11, 59, 369, 2665, 21823, 199983, 2028701, 22577141, 273551115, 3585133147, 50540288857, 762641865009, 12265883397719, 209475278413895, 3785852926650453, 72191462591370733, 1448516763956727331, 30507960955933725171, 672958104387944656145

Offset

0,6

Comments

A000142(n-2) gives number of permutations with a 123 present.

It appears that a(n) = A180191(n-2) - A018934(n-3) for n>3.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Example

For n=5 we have 12345, 12354 and 41235, so a(5)=3.
For n=6 we have 123456, 123465, 123546, 123465, 123645, 123654, 412356, 451236, 512346, 541236 and 612354, so a(6)=11.

Maple

a:= proc(n) option remember;
      `if`(n<7, [0$3, 1$2, 3, 11][n+1],
       ((4*n^3-42*n^2+92*n+39) *a(n-1)
        +(32*n^3-2*n^4-163*n^2+223*n+204) *a(n-2)
        -(n-4)*(n-7)*(2*n^2-10*n-15) *a(n-3)) / (2*n^2-14*n-3))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 07 2013

Mathematica

a[n_] := a[n] = If[n<7, {0, 0, 0, 1, 1, 3, 11}[[n+1]], ((4n^3 - 42n^2 + 92n + 39) a[n-1] + (32n^3 - 2n^4 - 163n^2 + 223n + 204) a[n-2] - (n-4)(n-7) (2n^2 - 10n - 15) a[n-3])/(2n^2 - 14n - 3)];
a /@ Range[0, 30] (* Jean-François Alcover, Mar 15 2021, after Alois P. Heinz *)

Crossrefs

Cf. A000142, A002467, A018934.

Sequence in context: A156560 A028342 A137982 * A074509 A343623 A282705

Adjacent sequences: A193361 A193362 A193363 * A193365 A193366 A193367

Keyword

nonn

Author

Jon Perry, Dec 20 2012

Status

approved