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