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

%I A145222 #22 Jan 31 2025 03:56:27
%S A145222 0,0,3,0,30,120,945,7392,66780,667440,7342335,88107360,1145396538,
%T A145222 16035550440,240533257965,3848532125760,65425046139960,
%U A145222 1177650830516832,22375365779822715,447507315596450880,9397653627525472470,206748379805560389720,4755212735527888968873
%N A145222 a(n) is the number of odd permutations (of an n-set) with exactly 1 fixed point.
%H A145222 Paolo Xausa, <a href="/A145222/b145222.txt">Table of n, a(n) for n = 1..400</a>
%H A145222 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 A145222 a(n) = A145225(n,1) = n*A000387(n-1), (n > 0).
%F A145222 E.g.f.: x^3*exp(-x)/(2*(1-x)).
%F A145222 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 A145222 a(3) = 3 because there are exactly 3 odd permutations (of a 3-set) having 1 fixed point, namely: (12), (13), (23).
%t A145222 A145222[n_] := n*Subfactorial[n - 3]*Binomial[n - 1, 2]; Array[A145222, 25] (* _Paolo Xausa_, Jan 31 2025 *)
%o A145222 (PARI) x = 'x + O('x^30); Vec(serlaplace(((x^3)*exp(-x))/(2*(1-x)))) \\ _Michel Marcus_, Apr 04 2016
%Y A145222 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 A145222 nonn
%O A145222 1,3
%A A145222 _Abdullahi Umar_, Oct 09 2008
%E A145222 More terms from _Alois P. Heinz_, Apr 04 2016