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 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 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

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Last modified April 3 23:48 EDT 2020. Contains 333207 sequences. (Running on oeis4.)