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(1) <= 2, and p(2) <= 4.
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, ones on its three subdiagonals (with the exception of zeros in the (3,1), (4,1), and (5,2)-entries), and is zero elsewhere.
This is row 9 of Kløve's Table 3.
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 0..90
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.
R. Lagrange, Quelques résultats dans la métrique des permutations, Annales Scientifiques de l'École Normale Supérieure, Paris, 79 (1962), 199-241.
Index entries for linear recurrences with constant coefficients, signature (2,2,0,10,8,-2,-16,-10,-2,4,2,0,2,1).
FORMULA
G.f.: -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)). - Colin Barker, Dec 16 2014
MAPLE
with(LinearAlgebra):
A002528:= n-> `if` (n<=1, 0, Permanent (Matrix (n, (i, j)->
`if` (abs(j-i)<4 and [i, j]<>[3, 1] and [i, j]<>[4, 1] and [i, j]<>[5, 2], 1, 0)))):
MATHEMATICA
a[n_] := Permanent[Table[If[Abs[j - i] < 4 && {i, j} != {3, 1} && {i, j} != {4, 1} && {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[2 x^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 *)
LinearRecurrence[{2, 2, 0, 10, 8, -2, -16, -10, -2, 4, 2, 0, 2, 1}, {0, 0, 2, 4, 12, 32, 108, 336, 1036, 3120, 9540, 29244, 89768, 274788}, 20] (* Harvey P. Dale, Jan 04 2020 *)
PROG
(PARI) concat([0, 0], Vec(-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)) + O(x^100))) \\ Colin Barker, Dec 16 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Name and comments edited, and a(12)-a(28) from Nathaniel Johnston, Apr 10 2011
STATUS
approved