A145222 a(n) is the number of odd permutations (of an n-set) with exactly 1 fixed point.

0, 0, 3, 0, 30, 120, 945, 7392, 66780, 667440, 7342335, 88107360, 1145396538, 16035550440, 240533257965, 3848532125760, 65425046139960, 1177650830516832, 22375365779822715, 447507315596450880, 9397653627525472470, 206748379805560389720, 4755212735527888968873

Offset: 1

Author

Abdullahi Umar, Oct 09 2008

Keywords

• nonn

Examples

			a(3) = 3 because there are exactly 3 odd permutations (of a 3-set) having 1 fixed point, namely: (12), (13), (23).
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..400
• 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), A145219 (even permutations with exactly 1 fixed point), A145223 (odd permutations with exactly 2 fixed points).

Programs

Mathematica

A145222[n_] := n*Subfactorial[n - 3]*Binomial[n - 1, 2]; Array[A145222, 25] (* Paolo Xausa, Jan 31 2025 *)

PARI

x = 'x + O('x^30); Vec(serlaplace(((x^3)*exp(-x))/(2*(1-x)))) \\ Michel Marcus, Apr 04 2016

Formula

a(n) = A145225(n,1) = n*A000387(n-1), (n > 0).

E.g.f.: x^3*exp(-x)/(2*(1-x)).

D-finite with recurrence (-n+3)*a(n) +n*(n-4)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 06 2023

Extensions

More terms from Alois P. Heinz, Apr 04 2016