A145220 - OEIS

%I #13 Jul 06 2023 06:48:25

%S 1,0,0,20,45,504,3640,33480,333585,3671360,44053416,572698620,

%T 8017774765,120266629560,1924266062160,32712523070864,588825415257345,

%U 11187682889912640,223753657798223920,4698826813762738020,103374189902780192781,2377606367763944486840

%N a(n) is the number of even 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) = (n(n-1)/2)*A003221(n-2), (n > 1).

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

%F 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 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 (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 Cf. A003221, A145219, A145223.

%K nonn

%O 2,4

%A _Abdullahi Umar_, Oct 09 2008

%E More terms from _Alois P. Heinz_, Nov 19 2013