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 A007016 Number of permutations of length n with 1 fixed and 1 reflected point. (Formerly M4491) 36
 0, 1, 0, 0, 8, 20, 96, 656, 5568, 48912, 494080, 5383552, 65097600, 840566080, 11833898496, 176621049600, 2838024476672, 48060623405312, 868000333234176, 16441638519762944, 329723762151352320, 6907027877807330304 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Number of distinct solutions to the order n checkerboard problem, including symmetrical solutions: place n pieces on an n X n board so there is exactly one piece in each row, column and main diagonal. Compare A064280. Number of magic permutation matrices of order n. - Chai Wah Wu, Jan 15 2019 Upper bound for the number of diagonal transversals in a Latin square: A287647(n) <= A287648(n) <= a(n). - Eduard I. Vatutin, Jan 02 2020 REFERENCES Simpson, Todd; Permutations with unique fixed and reflected points. Ars Combin. 39 (1995), 97-108. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Jean-François Alcover, Table of n, a(n) for n = 0..100 F. Rakotondrajao, Magic squares, rook polynomials and permutations, Séminaire Lotharingien de Combinatoire, vol. 54A, article B54Ac, 2006. T. Simpson, Letter to N. J. A. Sloane, Mar. 1992 T. Simpson, Permutations with unique fixed and reflected points, Preprint. (Annotated scanned copy) E. Vatutin, Upper bound for the number of diagonal transversals in a Latin square (in Russian). FORMULA a(2*m) = m*(x(2*m) - (2*m-3)*x(2*m-1)), a(2*m+1) = (2*m+1)*x(2*m) + 3*m*x(2*m-1) - 2*m*(m-1)*x(2*m-2), where x(n) = A003471(n). MATHEMATICA x[n_] := x[n] = Integrate[If[EvenQ[n], (x^2 - 4*x + 2)^(n/2), (x - 1)*(x^2 - 4*x + 2)^((n - 1)/2)]/E^x, {x, 0, Infinity}]; a[n_ /; EvenQ[n]] := With[{m = n/2}, m*(x[2*m] - (2*m - 3)*x[2*m - 1])]; a[n_ /; OddQ[n]] := With[{m = (n - 1)/2}, (2*m + 1)*x[2*m] + 3*m*x[2*m - 1] - 2*m*(m - 1)*x[2*m - 2]]; Table[a[n], {n, 0, 21}] // Quiet (* Jean-François Alcover, Jun 29 2018 *) PROG (PARI) a(n) = {my(v = vector(n)); \\ v is A003471 for(n=4, length(v), v[n] = (n-1)*v[n-1] + 2*if(n%2==1, (n-1)*v[n-2], (n-2) * if(n==4, 1, v[n-4]))); if(n<4, [1, 0, 0][n], if(n%2==0, n*(v[n] - (n-3)*v[n-1]), 2*n*v[n-1] + 3*(n-1)*v[n-2] - (n-1)*(n-3)*v[n-3])/2)} \\ Andrew Howroyd, Sep 12 2017 CROSSREFS Cf. A003471, A064280. Sequence in context: A003685 A066011 A333156 * A129550 A215181 A014584 Adjacent sequences:  A007013 A007014 A007015 * A007017 A007018 A007019 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified August 4 10:11 EDT 2021. Contains 346447 sequences. (Running on oeis4.)