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

%I A000804 M5375 N2333 #38 Jun 24 2023 21:48:01
%S A000804 1,1,2,6,24,120,265,579,1265,2783,6208,13909,31337,70985,161545,
%T A000804 369024,845825,1944295,4480285,10345391,23930320,55435605,128577253,
%U A000804 298529333,693718721,1613210120,3753680073,8738534315,20351593033,47413960239,110493496000
%N A000804 Permanent of a certain cyclic n X n (0,1) matrix.
%C A000804 a(n) is the number of permutations of [ n ] allowing i->i+j (mod n), j=0..4.
%D A000804 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000804 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000804 Alois P. Heinz, <a href="/A000804/b000804.txt">Table of n, a(n) for n = 0..1000</a>
%H A000804 N. Metropolis et al., <a href="http://dx.doi.org/10.1016/S0021-9800(69)80058-X">Permanents of cyclic (0,1) matrices</a>, J. Combin. Theory, 7 (1969), 291-321.
%H A000804 H. Minc, <a href="http://dx.doi.org/10.4153/CMB-1964-023-3">Permanents of (0,1)-circulants</a>, Canad. Math. Bull., 7 (1964), 253-263.
%H A000804 <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>
%H A000804 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (3, 0, -2, -2, -6, 2, 4, 2, 2, -1, -1).
%F A000804 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).
%p A000804 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);
%t A000804 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_ *)
%Y A000804 Cf. A000805.
%Y A000804 Fifth column of triangle A008305. - _Vladeta Jovovic_, Oct 03 2003
%Y A000804 Cf. A260074. - _Alois P. Heinz_, Jul 14 2015
%K A000804 nonn
%O A000804 0,3
%A A000804 _N. J. A. Sloane_
%E A000804 More terms from _Vladeta Jovovic_, Oct 03 2003
%E A000804 Edited by _Alois P. Heinz_, Dec 18 2010