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%I #29 Dec 10 2014 01:59:02
%S 0,1,2,6,14,48,152,476,1425,4340,13288,40852,125124,382888,1171612,
%T 3587505,10985790,33638142,102988410,315318756,965432832,2955964296,
%U 9050522241,27710613432,84843476928,259771465608,795361704776,2435217884992
%N Number of permutations p on the set [n] with the properties that abs(p(i)-i) <= 3 for all i, p(1) <= 3, and p(4) >= 2.
%C a(n) is also the permanent of the n X n matrix that has ones on its diagonal, ones on its three superdiagonals (with the exception of a single zero in the (1,4)-entry), ones on its three subdiagonals (with the exception of a single zero in the (4,1)-entry), and is zero elsewhere.
%C This is row 5 of Kløve's Table 3.
%H Nathaniel Johnston, <a href="/A188491/b188491.txt">Table of n, a(n) for n = 0..100</a>
%H Torleiv Kløve, <a href="http://www.ii.uib.no/publikasjoner/texrap/pdf/2008-376.pdf"> Spheres of Permutations under the Infinity Norm - Permutations with limited displacement. </a> Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.
%F a(n) = A002526(n-1) + A002528(n-1) + A188494(n-1). - _Nathaniel Johnston_, Apr 08 2011
%F G.f.: -x*(x^3+x^2-1)*(x^3+2*x^2+x+1) / (x^13 +3*x^12 +3*x^11 +5*x^10 +9*x^9 +7*x^8 -3*x^7 -19*x^6 -21*x^5 -13*x^4 -3*x^3 -3*x^2 -x+1).
%p a:= n-> (Matrix(13, (i, j)-> `if`(i=j-1, 1, `if`(i=13, [-1, -3, -3, -5, -9, -7, 3, 19, 21, 13, 3, 3, 1][j], 0)))^n. <<0, 0, 1, (0$6), 1, 2, 6, 14>>)[9, 1]: seq(a(n), n=0..30); # _Alois P. Heinz_, Apr 08 2011
%t a[n_] := ((Table[Which[i == j-1, 1, i == 13, {-1, -3, -3, -5, -9, -7, 3, 19, 21, 13, 3, 3, 1}[[j]], True, 0], {i, 1, 13}, {j, 1, 13}] // MatrixPower[#, n]&).{0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 6, 14})[[9]]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 17 2014, after _Alois P. Heinz_ *)
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Apr 01 2011
%E Name and comments edited by _Nathaniel Johnston_, Apr 08 2011
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