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A323254
The determinant of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 2*n - 1, n - 1, ..., 1 and whose first column consists of 2*n - 1, 2*n - 2, ..., n.
6
1, 7, 58, 614, 8032, 125757, 2298208, 48075148, 1133554432, 29756555315, 860884417024, 27218972906226, 933850899349504, 34556209025624041, 1371957513591119872, 58174957356247084568, 2624017129323317493760, 125454378698728779884895, 6337442836338834419089408
OFFSET
1,2
COMMENTS
The trace of the matrix M(n) is A000384(n). [Corrected by Stefano Spezia, Dec 08 2019]
The sum of the first row of the matrix M(n) is A034856(n).
The sum of the first column of the matrix M(n) is A000326(n).
For n > 1, the sum of the superdiagonal of the matrix M(n) is A000290(n-1).
For n > 1, the sum of the subdiagonal of the matrix M(n) is A001105(n-1).
LINKS
Wikipedia, Toeplitz matrix
FORMULA
a(n) ~ (5*exp(1) + exp(-1)) * n^n / 4. - Vaclav Kotesovec, Jan 10 2019
EXAMPLE
For n = 1 the matrix M(1) is
1
with determinant Det(M(1)) = 1.
For n = 2 the matrix M(2) is
3, 1
2, 3
with Det(M(2)) = 7.
For n = 3 the matrix M(3) is
5, 2, 1
4, 5, 2
3, 4, 5
with Det(M(3)) = 58.
MATHEMATICA
b[i_]:=i; a[n_]:=Det[ToeplitzMatrix[Join[{b[2*n-1]}, Array[b, n-1, {2*n-2, n}]], Join[{b[2*n-1]}, Array[b, n-1, {n-1, 1}]]]]; Array[a, 20]
PROG
(PARI) tm(n) = {my(m = matrix(n, n, i, j, if (j==1, 2*n-i, n-j+1))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }
a(n) = matdet(tm(n)); \\ Stefano Spezia, Dec 11 2019
CROSSREFS
Cf. A323255 (permanent of matrix M(n)).
Sequence in context: A367321 A192940 A126130 * A123766 A377331 A362772
KEYWORD
nonn
AUTHOR
Stefano Spezia, Jan 09 2019
STATUS
approved