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A145223 a(n) is the number of odd permutations (of an n-set) with exactly 2 fixed points.

%I A145223 #25 Jan 30 2025 11:24:50
%S A145223 0,0,6,0,90,420,3780,33264,333900,3670920,44054010,572697840,
%T A145223 8017775766,120266628300,1924266063720,32712523068960,588825415259640,
%U A145223 11187682889909904,223753657798227150,4698826813762734240,103374189902780197170,2377606367763944481780
%N A145223 a(n) is the number of odd permutations (of an n-set) with exactly 2 fixed points.
%H A145223 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 A145223 a(n) = A145225(n,2) = (n*(n-1)/2) * A000387(n-2), (n > 1).
%F A145223 E.g.f.: x^4*exp(-x)/(4*(1-x)).
%F A145223 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
%e A145223 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).
%p A145223 egf:= x^4 * exp(-x)/(4*(1-x));
%p A145223 a:= n-> n! * coeff(series(egf, x, n+1), x, n):
%p A145223 seq(a(n), n=2..30);  # _Alois P. Heinz_, Feb 01 2011
%t A145223 A000387[n_] := Subfactorial[n-2] Binomial[n, 2];
%t A145223 a[n_] := (n(n-1)/2) A000387[n-2];
%t A145223 Table[a[n], {n, 2, 30}] (* _Jean-François Alcover_, Jan 30 2025 *)
%o A145223 (PARI) x = 'x + O('x^30); Vec(serlaplace(((x^4)*exp(-x))/(4*(1-x)))) \\ _Michel Marcus_, Apr 04 2016
%Y A145223 Cf. A000387 (odd permutations with no fixed points), A145222 (odd permutations with exactly 1 fixed point), A145220 (even permutations with exactly 2 fixed points).
%K A145223 nonn
%O A145223 2,3
%A A145223 _Abdullahi Umar_, Oct 09 2008
%E A145223 More terms from _Alois P. Heinz_, Feb 01 2011