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

1,3

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.

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

Example

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

Mathematica

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

Prog

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

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).

Sequence in context: A215586 A215680 A007415 * A377309 A058833 A368284

Adjacent sequences: A145219 A145220 A145221 * A145223 A145224 A145225

Keyword

nonn

Author

Abdullahi Umar, Oct 09 2008

Extensions

More terms from Alois P. Heinz, Apr 04 2016

Status

approved