A145219 - OEIS

%I #12 Jul 06 2023 06:42:50

%S 1,0,0,8,15,144,910,7440,66717,667520,7342236,88107480,1145396395,

%T 16035550608,240533257770,3848532125984,65425046139705,

%U 1177650830517120,22375365779822392,447507315596451240,9397653627525472071,206748379805560390160,4755212735527888968390

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

%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*A003221(n-1), (n > 0).

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

%F 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 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 (PARI) x = 'x + O('x^30); Vec(serlaplace((x*(1-x^2/2) * exp(-x))/(1-x))) \\ _Michel Marcus_, Apr 04 2016

%Y Cf. A003221, A145220, A145222.

%K nonn

%O 1,4

%A _Abdullahi Umar_, Oct 09 2008

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