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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved