A356492 a(n) is the determinant of a symmetric Toeplitz matrix M(n) whose first row consists of prime(n), prime(n-1), ..., prime(1).

1, 2, 5, 51, 264, 19532, -11904, 1261296, -2052864, 70621632, 24618221568, 3996020736, 743171562496, 24567175118848, -1257930752000, 864893030400, 12289833785344000, 1099483729459478528, 100515455071223808, 757166323365314560, 6294658173770137600, 7801939905505132544

Offset: 0

Author

Stefano Spezia, Aug 09 2022

Keywords

• sign

Comments

Conjecture: a(n) is prime only for n = 1 and 2.

Conjecture is true because a(n) is even for n >= 4. This is because all but two rows of the matrix consist of odd numbers. - Robert Israel, Oct 13 2023

Examples

			For n = 1 the matrix M(1) is
    2
with determinant a(1) = 2.
For n = 2 the matrix M(2) is
    3, 2
    2, 3
with determinant a(2) = 5.
For n = 3 the matrix M(3) is
    5, 3, 2
    3, 5, 3
    2, 3, 5
with determinant a(3) = 51.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..532
• Mathematics Stack Exchange, Determinant of a Toeplitz matrix
• Wikipedia, Toeplitz Matrix

Crossrefs

Cf. A033286 (trace of the matrix M(n)), A356484 (hafnian of the matrix M(2*n)), A356493 (permanent of the matrix M(n)).

Cf. A350955, A350956.

Programs

Maple

f:=proc(n) uses LinearAlgebra; local i;
 Determinant(ToeplitzMatrix([seq(ithprime(i),i=n..1,-1)],symmetric));
end proc:
q(0):= 1:
map(q, [$0..25]); # Robert Israel, Oct 13 2023

Mathematica

k[i_]:=Prime[i]; M[ n_]:=ToeplitzMatrix[Reverse[Array[k, n]]]; a[n_]:=Det[M[n]]; Join[{1},Table[a[n],{n,21}]]

PARI

a(n) = matdet(apply(prime, matrix(n,n,i,j,n-abs(i-j)))); \\ Michel Marcus, Aug 12 2022

Formula

A350955(n) <= a(n) <= A350956(n).