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A145219 a(n) is the number of even permutations (of an n-set) with exactly 1 fixed point.

%I A145219 #12 Jul 06 2023 06:42:50
%S A145219 1,0,0,8,15,144,910,7440,66717,667520,7342236,88107480,1145396395,
%T A145219 16035550608,240533257770,3848532125984,65425046139705,
%U A145219 1177650830517120,22375365779822392,447507315596451240,9397653627525472071,206748379805560390160,4755212735527888968390
%N A145219 a(n) is the number of even permutations (of an n-set) with exactly 1 fixed point.
%H A145219 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 A145219 a(n) = n*A003221(n-1), (n > 0).
%F A145219 E.g.f.: (x*(1-x^2/2) * exp(-x))/(1-x).
%F A145219 D-finite with recurrence (-3*n+7)*a(n) +(n-2)*(3*n-10)*a(n-1) +(9*n-8)*(n-3)*a(n-2) +2*(3*n-2)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Jul 06 2023
%e A145219 a(4) = 8 because there are exactly 8 even permutations (of a 4-set) having 1 fixed point, namely: (123), (132), (124), (142), (134), (143), (234), (243).
%o A145219 (PARI) x = 'x + O('x^30); Vec(serlaplace((x*(1-x^2/2) * exp(-x))/(1-x))) \\ _Michel Marcus_, Apr 04 2016
%Y A145219 Cf. A003221, A145220, A145222.
%K A145219 nonn
%O A145219 1,4
%A A145219 _Abdullahi Umar_, Oct 09 2008
%E A145219 More terms from _Alois P. Heinz_, Nov 19 2013