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A145220 a(n) is the number of even permutations (of an n-set) with exactly 2 fixed points.

%I A145220 #13 Jul 06 2023 06:48:25
%S A145220 1,0,0,20,45,504,3640,33480,333585,3671360,44053416,572698620,
%T A145220 8017774765,120266629560,1924266062160,32712523070864,588825415257345,
%U A145220 11187682889912640,223753657798223920,4698826813762738020,103374189902780192781,2377606367763944486840
%N A145220 a(n) is the number of even permutations (of an n-set) with exactly 2 fixed points.
%H A145220 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 A145220 a(n) = (n(n-1)/2)*A003221(n-2), (n > 1).
%F A145220 E.g.f.: (x^2*(1-x^2/2) * exp(-x))/(2*(1-x)).
%F A145220 D-finite with recurrence -(n-5)*(n-2)^2*a(n) +n*(n-3)*(n^2-7*n+8)*a(n-1) +n*(n-4)*(n-1)^2*a(n-2)=0. - _R. J. Mathar_, Jul 06 2023
%e A145220 a(5) = 20 because there are exactly 20 even permutations (of a 5-set) having 2 fixed points, namely: (123), (132), (124), (142), (125), (152), (134), (143), (135), (153), (145), (154), (234), (243), (235), (253), (245), (254), (345), (354).
%o A145220 (PARI) x = 'x + O('x^30); Vec(serlaplace((x^2*(1-x^2/2) * exp(-x))/(2*(1-x)))) \\ _Michel Marcus_, Apr 04 2016
%Y A145220 Cf. A003221, A145219, A145223.
%K A145220 nonn
%O A145220 2,4
%A A145220 _Abdullahi Umar_, Oct 09 2008
%E A145220 More terms from _Alois P. Heinz_, Nov 19 2013