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

Author

Abdullahi Umar, Oct 09 2008

Keywords

• nonn

Examples

			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).
		

Links

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

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).

Programs

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 *)

PARI

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

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

Extensions

More terms from Alois P. Heinz, Feb 01 2011