%I
%S 0,1,2,3,8,30,102,308,905,2744,8473,26112,79924,244204,747160,2288521,
%T 7009458,21461803,65704200,201162258,615922714,1885853660,5774072225,
%U 17678809840,54128358209,165728860112,507424764216,1553620027784,4756831354752
%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(j) >= 2 for j=3,4.
%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 zeros in the (1,3) and (1,4)entries), 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 13 of Kløve's Table 3.
%H Torleiv Kløve, <a href="http://www.ii.uib.no/publikasjoner/texrap/pdf/2008376.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 From _Nathaniel Johnston_, Apr 11 2011: (Start)
%F a(n) = A188497(n+1)  A188494(n).
%F a(n) = A002526(n1) + A002526(n2).
%F (End)
%F G.f.: (x^10 + 2*x^9 + x^8  2*x^6  2*x^5  2*x^4  3*x^3 + x) / (x^14 + 2*x^13 + 2*x^11 + 4*x^10  2*x^9  10*x^8  16*x^7  2*x^6 + 8*x^5 + 10*x^4 + 2*x^2 + 2*x  1).
%p with(LinearAlgebra):
%p A188498:= n> `if` (n=0, 0, Permanent (Matrix (n, (i, j)>
%p `if` (abs(ji)<4 and [i, j]<>[1, 3] and [i, j]<>[1, 4] and [i, j]<>[3, 1] and [i, j]<>[4, 1], 1, 0)))):
%p seq (A188498(n), n=0..20);
%t a[n_] := Permanent[Table[If[Abs[j  i] < 4 && {i, j} != {1, 3} && {i, j} != {1, 4} && {i, j} != {3, 1} && {i, j} != {4, 1}, 1, 0], {i, 1, n}, {j, 1, n}]]; a[1] = 1; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 20}] (* _JeanFrançois Alcover_, Jan 07 2016, adapted from Maple *)
%t CoefficientList[Series[(x^10 + 2 x^9 + x^8  2 x^6  2 x^5  2 x^4  3 x^3 + x) / (x^14 + 2 x^13 + 2 x^11 + 4 x^10  2 x^9  10 x^8  16 x^7  2 x^6 + 8 x^5 + 10 x^4 + 2 x^2 + 2 x  1), {x, 0, 33}], x] (* _Vincenzo Librandi_, Jan 07 2016 *)
%o (PARI) concat(0, Vec((x^10+2*x^9+x^8 2*x^62*x^52*x^4 3*x^3+x) / (x^14+2*x^13+2*x^11 +4*x^102*x^910*x^8 16*x^72*x^6+8*x^5 +10*x^4+2*x^2+2*x1) + O(x^40))) \\ _Michel Marcus_, Dec 12 2014
%K nonn
%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 11 2011
