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A350549 a(n) is the permanent of a square matrix M(n) whose general element M_{i,j} is defined by floor((j - i + 1)/2). 2
1, 0, 0, -1, 2, 20, -120, -4608, 41952, 2325024, -34876800, -3133087200, 66120252480, 8258565859200, -239533775631360, -40631838221721600, 1532513262269767680, 335620705700380262400, -16054693916748370329600, -4428138916386119015424000, 261291002534430572648448000 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
The matrix M(n) is the n-th principal submatrix of the array A010751.
In the n X n matrix M(n): the zero element appears 2*n - 1 times; the positive integers k appears iff 0 < k < floor(n/2), 2*n - 1 - A040002(k-1) times; the negative integer k appears iff -k < ceiling(n/2), 2*n - 5 + 4*(k + 1) times.
det(M(n)) = 0, except for n = 3 for which det(M(3)) = -1.
The trace and the subdiagonal sum of the matrix M(n) are zero.
The antitrace of the matrix M(n) is A142150(n+1).
The superdiagonal sum of the matrix M(n) is equal to n - 1.
The sum of the elements of the matrix M(n) is A002620(n).
LINKS
EXAMPLE
For n = 3 the matrix M(3) is
0, 1, 1
0, 0, 1
-1, 0, 0
with permanent a(3) = -1.
For n = 4 the matrix M(4) is
0, 1, 1, 2
0, 0, 1, 1
-1, 0, 0, 1
-1, -1, 0, 0
with permanent a(4) = 2.
MAPLE
a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](
Matrix(n, (i, j)-> floor((j-i+1)/2)))):
seq(a(n), n=0..20); # Alois P. Heinz, Jan 19 2022
MATHEMATICA
Join[{1}, Table[Permanent[Table[Floor[(j-i+1)/2], {i, n}, {j, n}]], {n, 20}]]
PROG
(PARI) a(n) = matpermanent(matrix(n, n, i, j, (j - i + 1)\2)); \\ Michel Marcus, Jan 04 2022
(Python)
from sympy import Matrix
def A350549(n): return 1 if n == 0 else Matrix(n, n, lambda i, j:(j-i+1)//2).per() # Chai Wah Wu, Jan 12 2022
CROSSREFS
Sequence in context: A196797 A196585 A197042 * A062189 A203605 A094254
KEYWORD
sign
AUTHOR
Stefano Spezia, Jan 04 2022
STATUS
approved

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Last modified March 1 08:00 EST 2024. Contains 370430 sequences. (Running on oeis4.)