A356491 - OEIS

%I #25 Jan 31 2023 05:36:23

%S 1,2,13,184,4745,215442,14998965,1522204560,208682406913,

%T 37467772675962,8809394996942597,2597094620811897948,

%U 954601857873086235553,428809643170145564168434,229499307540038336275308821,144367721963876506217872778284,106064861375232790889279725340713

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

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

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

%e For n = 1 the matrix M(1) is

%e     2

%e with permanent a(1) = 2.

%e For n = 2 the matrix M(2) is

%e     2, 3

%e     3, 2

%e with permanent a(2) = 13.

%e For n = 3 the matrix M(3) is

%e     2, 3, 5

%e     3, 2, 3

%e     5, 3, 2

%e with permanent a(3) = 184.

%p A356491 := proc(n)

%p     local c ;

%p     if n =0 then

%p         return 1 ;

%p     end if;

%p     LinearAlgebra[ToeplitzMatrix]([seq(ithprime(c),c=1..n)],n,symmetric) ;

%p     LinearAlgebra[Permanent](%) ;

%p end proc:

%p seq(A356491(n),n=0..15) ; # _R. J. Mathar_, Jan 31 2023

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

%o (PARI) a(n) = matpermanent(apply(prime, matrix(n,n,i,j,abs(i-j)+1))); \\ _Michel Marcus_, Aug 12 2022

%o (Python)

%o from sympy import Matrix, prime

%o 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 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 Cf. A351021, A351022.

%K nonn

%O 0,2

%A _Stefano Spezia_, Aug 09 2022