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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).

%I A356492 #20 Oct 13 2023 12:28:14
%S A356492 1,2,5,51,264,19532,-11904,1261296,-2052864,70621632,24618221568,
%T A356492 3996020736,743171562496,24567175118848,-1257930752000,864893030400,
%U A356492 12289833785344000,1099483729459478528,100515455071223808,757166323365314560,6294658173770137600,7801939905505132544
%N 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).
%C A356492 Conjecture: a(n) is prime only for n = 1 and 2.
%C A356492 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
%H A356492 Robert Israel, <a href="/A356492/b356492.txt">Table of n, a(n) for n = 0..532</a>
%H A356492 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/3736861/determinant-of-a-toeplitz-matrix">Determinant of a Toeplitz matrix</a>
%H A356492 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>
%F A356492 A350955(n) <= a(n) <= A350956(n).
%e A356492 For n = 1 the matrix M(1) is
%e A356492     2
%e A356492 with determinant a(1) = 2.
%e A356492 For n = 2 the matrix M(2) is
%e A356492     3, 2
%e A356492     2, 3
%e A356492 with determinant a(2) = 5.
%e A356492 For n = 3 the matrix M(3) is
%e A356492     5, 3, 2
%e A356492     3, 5, 3
%e A356492     2, 3, 5
%e A356492 with determinant a(3) = 51.
%p A356492 f:=proc(n) uses LinearAlgebra; local i;
%p A356492  Determinant(ToeplitzMatrix([seq(ithprime(i),i=n..1,-1)],symmetric));
%p A356492 end proc:
%p A356492 q(0):= 1:
%p A356492 map(q, [$0..25]); # _Robert Israel_, Oct 13 2023
%t A356492 k[i_]:=Prime[i]; M[ n_]:=ToeplitzMatrix[Reverse[Array[k, n]]]; a[n_]:=Det[M[n]]; Join[{1},Table[a[n],{n,21}]]
%o A356492 (PARI) a(n) = matdet(apply(prime, matrix(n,n,i,j,n-abs(i-j)))); \\ _Michel Marcus_, Aug 12 2022
%Y A356492 Cf. A033286 (trace of the matrix M(n)), A356484 (hafnian of the matrix M(2*n)), A356493 (permanent of the matrix M(n)).
%Y A356492 Cf. A350955, A350956.
%K A356492 sign
%O A356492 0,2
%A A356492 _Stefano Spezia_, Aug 09 2022