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

1, 2, 13, 184, 4745, 215442, 14998965, 1522204560, 208682406913, 37467772675962, 8809394996942597, 2597094620811897948, 954601857873086235553, 428809643170145564168434, 229499307540038336275308821, 144367721963876506217872778284, 106064861375232790889279725340713

Offset: 0

Author

Stefano Spezia, Aug 09 2022

Keywords

• nonn

Comments

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

Examples

			For n = 1 the matrix M(1) is
    2
with permanent a(1) = 2.
For n = 2 the matrix M(2) is
    2, 3
    3, 2
with permanent a(2) = 13.
For n = 3 the matrix M(3) is
    2, 3, 5
    3, 2, 3
    5, 3, 2
with permanent a(3) = 184.
		

Links

• Mathematics Stack Exchange, Determinant of a Toeplitz matrix
• Wikipedia, Toeplitz Matrix

Crossrefs

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)).

Cf. A351021, A351022.

Programs

Maple

A356491 := proc(n)
    local c ;
    if n =0 then
        return 1 ;
    end if;
    LinearAlgebra[ToeplitzMatrix]([seq(ithprime(c),c=1..n)],n,symmetric) ;
    LinearAlgebra[Permanent](%) ;
end proc:
seq(A356491(n),n=0..15) ; # R. J. Mathar, Jan 31 2023

Mathematica

k[i_]:=Prime[i]; M[ n_]:=ToeplitzMatrix[Array[k, n]]; a[n_]:=Permanent[M[n]]; Join[{1},Table[a[n],{n,16}]]

PARI

a(n) = matpermanent(apply(prime, matrix(n,n,i,j,abs(i-j)+1))); \\ Michel Marcus, Aug 12 2022

Python

from sympy import Matrix, prime
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

Formula

A351021(n) <= a(n) <= A351022(n).