A322908 The determinant of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 1, ..., n and whose first column consists of 1, n + 1, ..., 2*n - 1.

1, -5, 38, -386, 4928, -75927, 1371808, -28452356, 666445568, -17402398505, 501297595904, -15792876550662, 540190822408192, -19937252888438459, 789770307546718208, -33422580292067020808, 1504926927960887066624, -71839548181524098808909, 3624029163661165580910592

Offset: 1

Author

Stefano Spezia, Dec 30 2018

Keywords

• sign

Comments

The matrix M(n) differs from that of A318173 in using successive positive integers in place of successive prime numbers.

The trace of the matrix M(n) is A000027(n).

The sum of the first row of the matrix M(n) is A000217(n).

The sum of the first column of the matrix M(n) is A005448(n). [Corrected by Stefano Spezia, Dec 11 2019]

For n > 1, the sum of the superdiagonal of the matrix M(n) is A005843(n).

Examples

			For n = 1 the matrix M(1) is
   1
with determinant Det(M(1)) = 1.
For n = 2 the matrix M(2) is
   1, 2
   3, 1
with Det(M(2)) = -5.
For n = 3 the matrix M(3) is
   1, 2, 3
   4, 1, 2
   5, 4, 1
with Det(M(3)) = 38.
		

Links

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

Crossrefs

Cf. A000027, A000217, A005448, A005843, A318173.

Cf. A322909 (permanent of matrix M(n)).

Programs

Maple

a:= proc(n) uses LinearAlgebra;
Determinant(ToeplitzMatrix([seq(i, i=2*n-1..n+1, -1), seq(i, i=1..n)]))
end proc:
map(a, [$1..20]);

Mathematica

b[n_]:=n; a[n_]:=Det[ToeplitzMatrix[Join[{b[1]}, Array[b, n-1, {n+1, 2*n-1}]], Array[b, n]]]; Array[a, 20]

PARI

tm(n) = {my(m = matrix(n, n, i, j, if (i==1, j, if (j==1, n+i-1)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }
a(n) = matdet(tm(n)); \\ Michel Marcus, Nov 11 2020

Formula

a(n) ~ -(-1)^n * (3*exp(1) - exp(-1)) * n^n / 4. - Vaclav Kotesovec, Jan 05 2019