

A161124


Number of inversions in all fixedpointfree involutions of {1,2,...,2n}.


5



0, 1, 12, 135, 1680, 23625, 374220, 6621615, 129729600, 2791213425, 65472907500, 1663666579575, 45537716624400, 1336089255125625, 41837777148667500, 1392813754566609375, 49126088694402720000, 1830138702650463830625, 71812362934450726087500
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

Also the sum of the major indices of all fixedpointfree involutions of {1,2,...,2n}. Example: a(2)=12 because the fixedpointfree 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*(2n1)!!.
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
Dfinite with recurrence a(n) +(2*n7)*a(n1) +(8*n3)*a(n2) +(2*n+5)*a(n3)=0.  R. J. Mathar, Jul 26 2022


EXAMPLE

a(2) = 12 because the fixedpointfree 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 (2n1)!!, {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



