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.

1, 7, 58, 614, 8032, 125757, 2298208, 48075148, 1133554432, 29756555315, 860884417024, 27218972906226, 933850899349504, 34556209025624041, 1371957513591119872, 58174957356247084568, 2624017129323317493760, 125454378698728779884895, 6337442836338834419089408

Offset: 1

Author

Stefano Spezia, Jan 09 2019

Keywords

• nonn

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).

Examples

			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.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..300
• Wikipedia, Toeplitz matrix

Crossrefs

Cf. A000290, A000326, A000384, A001105, A001792.

Cf. A034856, A204235, A318173, A322908, A322909.

Cf. A323255 (permanent of matrix M(n)).

Programs

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]

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

Formula

a(n) ~ (5*exp(1) + exp(-1)) * n^n / 4. - Vaclav Kotesovec, Jan 10 2019