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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.
8
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
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. A322909 (permanent of matrix M(n)).
Sequence in context: A243690 A335530 A308877 * A098937 A190314 A360349
KEYWORD
sign
AUTHOR
Stefano Spezia, Dec 30 2018
STATUS
approved