login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A161124 Number of inversions in all fixed-point-free 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 9 19:36 EST 2022. Contains 358703 sequences. (Running on oeis4.)