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Determinant of n X n matrix A[i,j] = (j - i - 1 mod n) + [i=j], i.e., the circulant having (n, 0, 1, ..., n-2) as first row.
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%I #19 Oct 18 2021 18:29:23

%S 1,1,4,28,273,3421,52288,941578,19505545,456790123,11931215316,

%T 343871642632,10840081272265,371026432467913,13702802011918048,

%U 543154131059225686,23000016472483168305,1036227971225610466711,49492629462587441963140,2497992686980609418282548,132849300060919364474261281

%N Determinant of n X n matrix A[i,j] = (j - i - 1 mod n) + [i=j], i.e., the circulant having (n, 0, 1, ..., n-2) as first row.

%C It is remarkable that for odd n, this determinant has its base n+1 digits equal to the middle row: e.g., a(9) = 456790123 is the determinant of the circulant matrix having [4,5,6,7,9,0,1,2,3] as middle row.

%C a(0) = 1 is (by convention) the determinant of a 0 X 0 matrix.

%H Max Alekseyev, <a href="/A219324/a219324.png">Illustration for a(9) = 456790123 = A219324(20)</a>.

%H N. I. Belukhov, <a href="http://www.mccme.ru/free-books/matpros/mpf.pdf">Solution to Problem 14.7</a> (in Russian), Matematicheskoe Prosveshchenie 15 (2011), pp. 241-244.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Circulant_matrix">Circulant matrix</a>.

%F a(n) = det(I(n) + C(n)), where I(n) is the n X n identity matrix and C(n) is the circulant having (n-1, ..., 0) as first column.

%e a(5) = 3421 is the determinant of the matrix

%e ( 5 0 1 2 3 )

%e ( 3 5 0 1 2 )

%e ( 2 3 5 0 1 ) and 3421 = 23501[6], i.e., written in base 6.

%e ( 1 2 3 5 0 )

%e ( 0 1 2 3 5 ).

%o (PARI) a(n)=matdet(matrix(n,n,i,j,(j-i-1)%n+(i==j)))

%o (Python)

%o from sympy import Matrix

%o def A303260(n): return Matrix(n,n, lambda i,j:(j-i-1) % n + (i==j)).det() # _Chai Wah Wu_, Oct 18 2021

%Y Cf. A081131(n+1) = determinant of the circulant matrix C(n) defined in formula, A070896 (signed variant).

%Y See also A219324.

%K nonn

%O 0,3

%A _M. F. Hasler_, Apr 23 2018