A145223 - OEIS

%I #25 Jan 30 2025 11:24:50

%S 0,0,6,0,90,420,3780,33264,333900,3670920,44054010,572697840,

%T 8017775766,120266628300,1924266063720,32712523068960,588825415259640,

%U 11187682889909904,223753657798227150,4698826813762734240,103374189902780197170,2377606367763944481780

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

%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,2) = (n*(n-1)/2) * A000387(n-2), (n > 1).

%F E.g.f.: x^4*exp(-x)/(4*(1-x)).

%F 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 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 egf:= x^4 * exp(-x)/(4*(1-x));

%p a:= n-> n! * coeff(series(egf, x, n+1), x, n):

%p seq(a(n), n=2..30);  # _Alois P. Heinz_, Feb 01 2011

%t A000387[n_] := Subfactorial[n-2] Binomial[n, 2];

%t a[n_] := (n(n-1)/2) A000387[n-2];

%t Table[a[n], {n, 2, 30}] (* _Jean-François Alcover_, Jan 30 2025 *)

%o (PARI) x = 'x + O('x^30); Vec(serlaplace(((x^4)*exp(-x))/(4*(1-x)))) \\ _Michel Marcus_, Apr 04 2016

%Y 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 nonn

%O 2,3

%A _Abdullahi Umar_, Oct 09 2008

%E More terms from _Alois P. Heinz_, Feb 01 2011