%I #29 Jul 26 2022 15:52:47

%S 0,1,12,135,1680,23625,374220,6621615,129729600,2791213425,

%T 65472907500,1663666579575,45537716624400,1336089255125625,

%U 41837777148667500,1392813754566609375,49126088694402720000,1830138702650463830625,71812362934450726087500

%N Number of inversions in all fixed-point-free involutions of {1,2,...,2n}.

%C Also the sum of the major indices of all fixed-point-free involutions of {1,2,...,2n}. Example: a(2)=12 because the fixed-point-free involutions 2143, 3412, and 4321 have major indices 4, 2, and 6, respectively.

%C a(n) = Sum(k*A161123(n,k), k>=0).

%C For n > 0, a(n) is also the determinant absolute value of the symmetric n X n matrix M defined by M(i,j) = max(i,j)^2 for 1 <= i,j <= n. - _Enrique Pérez Herrero_, Jan 14 2013

%H Alois P. Heinz, <a href="/A161124/b161124.txt">Table of n, a(n) for n = 0..200</a>

%H E. Pérez Herrero, <a href="http://psychedelic-geometry.blogspot.com.es/2013/01/max-determinant.html">Max Determinant</a>, Psychedelic Geometry Blogspot, 15 Jan 2013

%F a(n) = n^2*(2n-1)!!.

%F a(n) = n^2*A001147(n). - _Enrique Pérez Herrero_, Jan 14 2013

%F a(n) = (2n)! * [x^(2n)] (x^2/2 + x^4/4)*exp(x^2/2). - _Geoffrey Critzer_, Mar 03 2013

%F D-finite with recurrence a(n) +(-2*n-7)*a(n-1) +(8*n-3)*a(n-2) +(-2*n+5)*a(n-3)=0. - _R. J. Mathar_, Jul 26 2022

%e a(2) = 12 because the fixed-point-free involutions 2143, 3412, and 4321 have 2, 4, and 6 inversions, respectively.

%p seq(n^2*factorial(2*n)/(factorial(n)*2^n), n = 0 .. 18);

%t nn=40;Prepend[Select[Range[0,nn]!CoefficientList[Series[(x^2/2+x^4/4)Exp[x^2/2],{x,0,nn}],x],#>0&],0] (* _Geoffrey Critzer_, Mar 03 2013 *)

%t Table[n^2 (2n-1)!!,{n,0,20}] (* _Harvey P. Dale_, Jan 05 2014 *)

%Y Cf. A161123.

%Y Cf. A051125, A181983, A211606.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Jun 05 2009