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%I #21 Jan 06 2016 13:53:40
%S 0,1,2,4,10,36,120,368,1089,3304,10168,31312,95880,293120,896824,
%T 2746569,8411818,25756220,78853410,241421436,739183568,2263249600,
%U 6929580817,21216729488,64960656448,198894856144,608971496032,1864533223584,5708777321872
%N Number of permutations p on the set [n] with the properties that abs(p(i)-i) <= 3 for all i, p(1) <= 2, 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 zero in the (1,4)-entry), ones on its three subdiagonals (with the exception of zeros in the (3,1) and (4,1)-entries), and is zero elsewhere.
%C This is row 10 of Kløve's Table 3.
%H Nathaniel Johnston, <a href="/A188495/b188495.txt">Table of n, a(n) for n = 0..119</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.
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,0,10,8,-2,-16,-10,-2,4,2,0,2,1).
%F From _Nathaniel Johnston_, Apr 10 2011: (Start)
%F a(n) = A188493(n+1) - A188491(n) - A188497(n).
%F a(n) = A002526(n-1) + A188494(n-1).
%F (End)
%F G.f.: -x*(x +1)*(x^6 +x^5 -x^4 -x^3 -x^2 -x +1) / ((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)). - _Colin Barker_, Dec 13 2014
%p with (LinearAlgebra):
%p A188495:= n-> `if` (n=0, 0, Permanent (Matrix (n, (i, j)->
%p `if` (abs(j-i)<4 and [i, j]<>[3, 1] and [i, j]<>[4, 1] and [i, j]<>[1, 4], 1, 0)))):
%p seq (A188495(n), n=0..20);
%t a[n_] := Permanent[Table[If[Abs[j - i] < 4 && {i, j} != {3, 1} && {i, j} != {4, 1} && {i, j} != {1, 4}, 1, 0], {i, 1, n}, {j, 1, n}]]; a[1] = 1; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 20}] (* _Jean-François Alcover_, Jan 06 2016, adapted from Maple *)
%o (PARI) concat(0, Vec(-x*(x +1)*(x^6 +x^5 -x^4 -x^3 -x^2 -x +1) / ((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)) + O(x^100))) \\ _Colin Barker_, Dec 13 2014
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Apr 01 2011
%E Name and comments edited, and a(12)-a(28) from _Nathaniel Johnston_, Apr 10 2011
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