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a(n) is the number of odd permutations (of an n-set) with exactly 1 fixed point.
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%I #17 Jul 06 2023 06:39:56

%S 0,0,3,0,30,120,945,7392,66780,667440,7342335,88107360,1145396538,

%T 16035550440,240533257965,3848532125760,65425046139960,

%U 1177650830516832,22375365779822715,447507315596450880,9397653627525472470,206748379805560389720,4755212735527888968873

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

%H Bashir Ali and A. Umar, <a href="http://www.seams-bull-math.ynu.edu.cn/downloadfile.jsp?filemenu=_200805&amp;filename=Some Combinatorial Properties of the Alternating Group.pdf">Some combinatorial properties of the alternating group</a>, Southeast Asian Bulletin Math. 32 (2008), 823-830.

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

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

%F 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

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

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

%Y Cf. A000387 (odd permutations with no fixed points), A145219 (even permutations with exactly 1 fixed point), A145223 (odd permutations with exactly 2 fixed points).

%K nonn

%O 1,3

%A _Abdullahi Umar_, Oct 09 2008

%E More terms from _Alois P. Heinz_, Apr 04 2016