A145223 a(n) is the number of odd permutations (of an n-set) with exactly 2 fixed points.

0, 0, 6, 0, 90, 420, 3780, 33264, 333900, 3670920, 44054010, 572697840, 8017775766, 120266628300, 1924266063720, 32712523068960, 588825415259640, 11187682889909904, 223753657798227150, 4698826813762734240, 103374189902780197170, 2377606367763944481780

Offset

2,3

Links

Table of n, a(n) for n=2..23.

Bashir Ali and A. Umar, Some combinatorial properties of the alternating group, Southeast Asian Bulletin Math. 32 (2008), 823-830.

Formula

a(n) = A145225(n,2) = (n*(n-1)/2) * A000387(n-2), (n > 1).

E.g.f.: x^4*exp(-x)/(4*(1-x)).

D-finite with recurrence +(-n+6)*a(n) +(n-2)*(n-7)*a(n-1) +(n-2)*(n-3)*a(n-2)=0. - R. J. Mathar, Jul 06 2023

Example

a(4) = 6 because there are exactly 6 odd permutations (of a 4-set) having 2 fixed points, namely: (12), (13), (14), (23), (24), (34).

Maple

egf:= x^4 * exp(-x)/(4*(1-x));
a:= n-> n! * coeff(series(egf, x, n+1), x, n):
seq(a(n), n=2..30);  # Alois P. Heinz, Feb 01 2011

Mathematica

A000387[n_] := Subfactorial[n-2] Binomial[n, 2];
a[n_] := (n(n-1)/2) A000387[n-2];
Table[a[n], {n, 2, 30}] (* Jean-François Alcover, Jan 30 2025 *)

Prog

(PARI) x = 'x + O('x^30); Vec(serlaplace(((x^4)*exp(-x))/(4*(1-x)))) \\ Michel Marcus, Apr 04 2016

Crossrefs

Cf. A000387 (odd permutations with no fixed points), A145222 (odd permutations with exactly 1 fixed point), A145220 (even permutations with exactly 2 fixed points).

Sequence in context: A057399 A245086 A365909 * A365979 A219948 A072129

Adjacent sequences: A145220 A145221 A145222 * A145224 A145225 A145226

Keyword

nonn

Author

Abdullahi Umar, Oct 09 2008

Extensions

More terms from Alois P. Heinz, Feb 01 2011

Status

approved