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A356491 a(n) is the permanent of a symmetric Toeplitz matrix M(n) whose first row consists of prime(1), prime(2), ..., prime(n).

%I A356491 #25 Jan 31 2023 05:36:23
%S A356491 1,2,13,184,4745,215442,14998965,1522204560,208682406913,
%T A356491 37467772675962,8809394996942597,2597094620811897948,
%U A356491 954601857873086235553,428809643170145564168434,229499307540038336275308821,144367721963876506217872778284,106064861375232790889279725340713
%N A356491 a(n) is the permanent of a symmetric Toeplitz matrix M(n) whose first row consists of prime(1), prime(2), ..., prime(n).
%C A356491 Conjecture: a(n) is prime only for n = 1 and 2.
%H A356491 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/3736861/determinant-of-a-toeplitz-matrix">Determinant of a Toeplitz matrix</a>
%H A356491 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>
%F A356491 A351021(n) <= a(n) <= A351022(n).
%e A356491 For n = 1 the matrix M(1) is
%e A356491     2
%e A356491 with permanent a(1) = 2.
%e A356491 For n = 2 the matrix M(2) is
%e A356491     2, 3
%e A356491     3, 2
%e A356491 with permanent a(2) = 13.
%e A356491 For n = 3 the matrix M(3) is
%e A356491     2, 3, 5
%e A356491     3, 2, 3
%e A356491     5, 3, 2
%e A356491 with permanent a(3) = 184.
%p A356491 A356491 := proc(n)
%p A356491     local c ;
%p A356491     if n =0 then
%p A356491         return 1 ;
%p A356491     end if;
%p A356491     LinearAlgebra[ToeplitzMatrix]([seq(ithprime(c),c=1..n)],n,symmetric) ;
%p A356491     LinearAlgebra[Permanent](%) ;
%p A356491 end proc:
%p A356491 seq(A356491(n),n=0..15) ; # _R. J. Mathar_, Jan 31 2023
%t A356491 k[i_]:=Prime[i]; M[ n_]:=ToeplitzMatrix[Array[k, n]]; a[n_]:=Permanent[M[n]]; Join[{1},Table[a[n],{n,16}]]
%o A356491 (PARI) a(n) = matpermanent(apply(prime, matrix(n,n,i,j,abs(i-j)+1))); \\ _Michel Marcus_, Aug 12 2022
%o A356491 (Python)
%o A356491 from sympy import Matrix, prime
%o A356491 def A356491(n): return Matrix(n,n,[prime(abs(i-j)+1) for i in range(n) for j in range(n)]).per() if n else 1 # _Chai Wah Wu_, Aug 12 2022
%Y A356491 Cf. A005843 (trace of the matrix M(n)), A309131 (k-superdiagonal sum of the matrix M(n)), A356483 (hafnian of the matrix M(2*n)), A356490 (determinant of the matrix M(n)).
%Y A356491 Cf. A351021, A351022.
%K A356491 nonn
%O A356491 0,2
%A A356491 _Stefano Spezia_, Aug 09 2022