A188497 a(n) = A188493(n+1) - A188491(n) - A188495(n).

0, 0, 2, 4, 7, 20, 72, 240, 722, 2140, 6508, 20077, 61776, 189056, 577856, 1768380, 5416230, 16587984, 50788707, 155489884, 476058864, 1457605616, 4462928950, 13664497400, 41837412392, 128096408137, 392202398144, 1200835918016, 3676688064688

Offset

0,3

Comments

For n >= 2, a(n) is the number of permutations p on the set [n] with the properties that abs(p(i)-i) <= 3 for all i, p(j) <= 1+j for j=1,2, and p(4) >= 2.

For n >= 2, 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), (4,1), (4,2), and (5,2)-entries), and is zero elsewhere.

This is row 12 of Kløve's Table 3.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Torleiv Kløve, Spheres of Permutations under the Infinity Norm - Permutations with limited displacement. Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.

Index entries for linear recurrences with constant coefficients, signature (2,2,0,10,8,-2,-16,-10,-2,4,2,0,2,1).

Formula

a(n) = A188494(n-1) + A188498(n-1). - Nathaniel Johnston, Apr 11 2011

G.f.: -x^2*(x^9 +2*x^8 -2*x^4 -2*x^3 -5*x^2 +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)). - Colin Barker, Dec 13 2014

Maple

with(LinearAlgebra):
A188497:= n-> `if` (n<=1, 0, Permanent (Matrix (n, (i, j)->
              `if` (abs(j-i)<4 and [i, j]<>[1, 4] and [i, j]<>[3, 1] and [i, j]<>[4, 1] and [i, j]<>[4, 2] and [i, j]<>[5, 2], 1, 0)))):
seq (A188497(n), n=0..20);

Mathematica

a[n_] := Permanent[Table[If[Abs[j - i] < 4 && {i, j} != {1, 4} && {i, j} != {3, 1} && {i, j} != {4, 1} && {i, j} != {4, 2} && {i, j} != {5, 2}, 1, 0], {i, 1, n}, {j, 1, n}]]; a[1] = 0; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 20}](* Jean-François Alcover, Jan 07 2016, adapted from Maple *)
CoefficientList[Series[x^2 (x^9 + 2 x^8 - 2 x^4 - 2 x^3 - 5 x^2 + 2) / ((1 - x) (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)), {x, 0, 33}], x] (* Vincenzo Librandi, Jan 07 2016

Prog

(PARI) concat([0, 0], Vec(-x^2*(x^9 +2*x^8 -2*x^4 -2*x^3 -5*x^2 +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)) + O(x^100))) \\ Colin Barker, Dec 13 2014

Crossrefs

Sequence in context: A291403 A101805 A145777 * A237759 A306978 A337062

Adjacent sequences: A188494 A188495 A188496 * A188498 A188499 A188500

Keyword

nonn,easy

Author

N. J. A. Sloane, Apr 01 2011

Extensions

Name and comments edited, and a(12)-a(28) from Nathaniel Johnston, Apr 11 2011

Status

approved