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

1,2

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).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..300

Wikipedia, Toeplitz Matrix

Formula

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

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]

Prog

(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

Crossrefs

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

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

Sequence in context: A243690 A335530 A308877 * A098937 A190314 A360349

Adjacent sequences: A322905 A322906 A322907 * A322909 A322910 A322911

Keyword

sign

Author

Stefano Spezia, Dec 30 2018

Status

approved