cp's OEIS Frontend

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).

%I A350549 #34 Jan 19 2022 12:27:29
%S A350549 1,0,0,-1,2,20,-120,-4608,41952,2325024,-34876800,-3133087200,
%T A350549 66120252480,8258565859200,-239533775631360,-40631838221721600,
%U A350549 1532513262269767680,335620705700380262400,-16054693916748370329600,-4428138916386119015424000,261291002534430572648448000
%N 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).
%C A350549 The matrix M(n) is the n-th principal submatrix of the array A010751.
%C A350549 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.
%C A350549 det(M(n)) = 0, except for n = 3 for which det(M(3)) = -1.
%C A350549 The trace and the subdiagonal sum of the matrix M(n) are zero.
%C A350549 The antitrace of the matrix M(n) is A142150(n+1).
%C A350549 The superdiagonal sum of the matrix M(n) is equal to n - 1.
%C A350549 The sum of the elements of the matrix M(n) is A002620(n).
%e A350549 For n = 3 the matrix M(3) is
%e A350549      0, 1, 1
%e A350549      0, 0, 1
%e A350549     -1, 0, 0
%e A350549 with permanent a(3) = -1.
%e A350549 For n = 4 the matrix M(4) is
%e A350549     0,  1,  1,  2
%e A350549     0,  0,  1,  1
%e A350549    -1,  0,  0,  1
%e A350549    -1, -1,  0,  0
%e A350549 with permanent a(4) = 2.
%p A350549 a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](
%p A350549          Matrix(n, (i, j)-> floor((j-i+1)/2)))):
%p A350549 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jan 19 2022
%t A350549 Join[{1},Table[Permanent[Table[Floor[(j-i+1)/2],{i,n},{j,n}]],{n,20}]]
%o A350549 (PARI) a(n) = matpermanent(matrix(n, n, i, j, (j - i + 1)\2)); \\ _Michel Marcus_, Jan 04 2022
%o A350549 (Python)
%o A350549 from sympy import Matrix
%o A350549 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
%Y A350549 Cf. A002620, A004526, A010751, A040002, A060747, A110654, A142150.
%K A350549 sign
%O A350549 0,5
%A A350549 _Stefano Spezia_, Jan 04 2022