A161124 Number of inversions in all fixed-point-free involutions of {1,2,...,2n}.

0, 1, 12, 135, 1680, 23625, 374220, 6621615, 129729600, 2791213425, 65472907500, 1663666579575, 45537716624400, 1336089255125625, 41837777148667500, 1392813754566609375, 49126088694402720000, 1830138702650463830625, 71812362934450726087500

Offset

0,3

Comments

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.

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

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

E. Pérez Herrero, Max Determinant, Psychedelic Geometry Blogspot, 15 Jan 2013

Formula

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

a(n) = n^2*A001147(n). - Enrique Pérez Herrero, Jan 14 2013

a(n) = (2n)! * [x^(2n)] (x^2/2 + x^4/4)*exp(x^2/2). - Geoffrey Critzer, Mar 03 2013

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

Example

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

Maple

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

Mathematica

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 *)
Table[n^2 (2n-1)!!, {n, 0, 20}] (* Harvey P. Dale, Jan 05 2014 *)

Crossrefs

Cf. A161123.

Cf. A051125, A181983, A211606.

Sequence in context: A218762 A199233 A085938 * A288035 A290474 A030023

Adjacent sequences: A161121 A161122 A161123 * A161125 A161126 A161127

Keyword

nonn

Author

Emeric Deutsch, Jun 05 2009

Status

approved