A000804 Permanent of a certain cyclic n X n (0,1) matrix. (Formerly M5375 N2333)

1, 1, 2, 6, 24, 120, 265, 579, 1265, 2783, 6208, 13909, 31337, 70985, 161545, 369024, 845825, 1944295, 4480285, 10345391, 23930320, 55435605, 128577253, 298529333, 693718721, 1613210120, 3753680073, 8738534315, 20351593033, 47413960239, 110493496000

Offset

0,3

Comments

a(n) is the number of permutations of [ n ] allowing i->i+j (mod n), j=0..4.

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

N. Metropolis et al., Permanents of cyclic (0,1) matrices, J. Combin. Theory, 7 (1969), 291-321.

H. Minc, Permanents of (0,1)-circulants, Canad. Math. Bull., 7 (1964), 253-263.

Index entries for sequences related to binary matrices

Index entries for linear recurrences with constant coefficients, signature (3, 0, -2, -2, -6, 2, 4, 2, 2, -1, -1).

Formula

G.f.: (41*x^15 +64*x^14 -48*x^13 -113*x^12 -213*x^11 -190*x^10 +122*x^9 +158*x^8 +150*x^7 +75*x^6 -60*x^5 -10*x^4 -2*x^3 +x^2 +2*x -1) / (-x^11 -x^10 +2*x^9 +2*x^8 +4*x^7 +2*x^6 -6*x^5 -2*x^4 -2*x^3 +3*x -1).

Maple

a:= n-> `if`(n<5, n!, (Matrix(11, (i, j)-> if i+1=j then 1 elif i=11 then [-1, -1, 2, 2, 4, 2, -6, -2, -2, 0, 3][j] else 0 fi)^(n+6). <<41, -16, 33, -1, 5, -1, 16, 5, 13, 29, 65>>)[1, 1]): seq(a(n), n=0..30);

Mathematica

a[n_] := If[n<5, n!, ((Table[Which[i+1 == j, 1, i == 11, {-1, -1, 2, 2, 4, 2, -6, -2, -2, 0, 3}[[j]], True, 0], {i, 1, 11}, {j, 1, 11}] // MatrixPower[#, n+6]&).{41, -16, 33, -1, 5, -1, 16, 5, 13, 29, 65}) // First]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)

Crossrefs

Cf. A000805.

Fifth column of triangle A008305. - Vladeta Jovovic, Oct 03 2003

Cf. A260074. - Alois P. Heinz, Jul 14 2015

Sequence in context: A263699 A083267 A130480 * A048631 A263700 A263701

Adjacent sequences: A000801 A000802 A000803 * A000805 A000806 A000807

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Oct 03 2003

Edited by Alois P. Heinz, Dec 18 2010

Status

approved