|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
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
|
|
|
FORMULA
|
|
|
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; }
|
|
CROSSREFS
|
Cf. A323255 (permanent of matrix M(n)).
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|