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

0,2

Comments

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

Links

Table of n, a(n) for n=0..16.

Mathematics Stack Exchange, Determinant of a Toeplitz matrix

Wikipedia, Toeplitz Matrix

Formula

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

Example

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.

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}]]

Prog

(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

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.

Sequence in context: A137610 A073178 A193192 * A226865 A062156 A371318

Adjacent sequences: A356488 A356489 A356490 * A356492 A356493 A356494

Keyword

nonn

Author

Stefano Spezia, Aug 09 2022

Status

approved