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A356493 a(n) is the permanent of a symmetric Toeplitz matrix M(n) whose first row consists of prime(n), prime(n-1), ..., prime(1).

%I A356493 #16 Oct 13 2023 11:50:34
%S A356493 1,2,13,271,12030,1346758,214022024,51763672608,16088934953136,
%T A356493 6611717516842608,4412314619046451200,3533754988232088933120,
%U A356493 3506189715435673999194112,4444138735439968822425464576,5893766827264238066914528545792,8502284313901016361834901076874240,15350799440394462109333953415858960384
%N A356493 a(n) is the permanent of a symmetric Toeplitz matrix M(n) whose first row consists of prime(n), prime(n-1), ..., prime(1).
%C A356493 Conjecture: a(n) is prime only for n = 1, 2, and 3.
%C A356493 Conjecture is true because a(n) is even for n >= 4. This is because a(n) == A356492(n) (mod 2), and all but two rows of the matrix consist of odd numbers. - _Robert Israel_, Oct 13 2023
%H A356493 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/3736861/determinant-of-a-toeplitz-matrix">Determinant of a Toeplitz matrix</a>
%H A356493 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>
%F A356493 A351021(n) <= a(n) <= A351022(n).
%e A356493 For n = 1 the matrix M(1) is
%e A356493     2
%e A356493 with permanent a(1) = 2.
%e A356493 For n = 2 the matrix M(2) is
%e A356493     3, 2
%e A356493     2, 3
%e A356493 with permanent a(2) = 13.
%e A356493 For n = 3 the matrix M(3) is
%e A356493     5, 3, 2
%e A356493     3, 5, 3
%e A356493     2, 3, 5
%e A356493 with permanent a(3) = 271.
%t A356493 k[i_]:=Prime[i]; M[ n_]:=ToeplitzMatrix[Reverse[Array[k, n]]]; a[n_]:=Permanent[M[n]]; PrimeQ[Join[{1},Table[a[n],{n,16}]]]
%o A356493 (PARI) a(n) = matpermanent(apply(prime, matrix(n,n,i,j,n-abs(i-j)))); \\ _Michel Marcus_, Aug 12 2022
%Y A356493 Cf. A033286 (trace of the matrix M(n)), A356484 (hafnian of the matrix M(2*n)), A356492 (determinant of the matrix M(n)).
%Y A356493 Cf. A351021, A351022.
%K A356493 nonn
%O A356493 0,2
%A A356493 _Stefano Spezia_, Aug 09 2022