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A356490 a(n) is the determinant of a symmetric Toeplitz matrix M(n) whose first row consists of prime(1), prime(2), ..., prime(n).

%I A356490 #25 Jan 31 2023 05:35:07
%S A356490 1,2,-5,12,-19,-22,1143,-4284,14265,-46726,-84405,1306096,32312445,
%T A356490 522174906,4105967871,5135940112,-642055973735,-2832632334858,
%U A356490 14310549077571,380891148658140,4888186898996125,-49513565563840210,383405170118692791,-2517836083641473036,-3043377347606882055
%N A356490 a(n) is the determinant of a symmetric Toeplitz matrix M(n) whose first row consists of prime(1), prime(2), ..., prime(n).
%C A356490 Conjecture: abs(a(n)) is prime only for n = 1, 2, and 4.
%H A356490 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/3736861/determinant-of-a-toeplitz-matrix">Determinant of a Toeplitz matrix</a>
%H A356490 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>
%F A356490 A350955(n) <= a(n) <= A350956(n).
%e A356490 For n = 1 the matrix M(1) is
%e A356490     2
%e A356490 with determinant a(1) = 2.
%e A356490 For n = 2 the matrix M(2) is
%e A356490     2, 3
%e A356490     3, 2
%e A356490 with determinant a(2) = -5.
%e A356490 For n = 3 the matrix M(3) is
%e A356490     2, 3, 5
%e A356490     3, 2, 3
%e A356490     5, 3, 2
%e A356490 with determinant a(3) = 12.
%p A356490 A356490 := proc(n)
%p A356490     local T,c ;
%p A356490     if n =0 then
%p A356490         return 1 ;
%p A356490     end if;
%p A356490     T := LinearAlgebra[ToeplitzMatrix]([seq(ithprime(c),c=1..n)],n,symmetric) ;
%p A356490     LinearAlgebra[Determinant](T) ;
%p A356490 end proc:
%p A356490 seq(A356490(n),n=0..15) ; # _R. J. Mathar_, Jan 31 2023
%t A356490 k[i_]:=Prime[i]; M[ n_]:=ToeplitzMatrix[Array[k, n]]; a[n_]:=Det[M[n]]; Join[{1},Table[a[n],{n,24}]]
%o A356490 (PARI) a(n) = matdet(apply(prime, matrix(n,n,i,j,abs(i-j)+1))); \\ _Michel Marcus_, Aug 12 2022
%o A356490 (Python)
%o A356490 from sympy import Matrix, prime
%o A356490 def A356490(n): return Matrix(n,n,[prime(abs(i-j)+1) for i in range(n) for j in range(n)]).det() # _Chai Wah Wu_, Aug 12 2022
%Y A356490 Cf. A005843 (trace of M(n)), A309131 (k-superdiagonal sum of M(n)), A356483 (hafnian of M(2*n)), A356491 (permanent of M(n)).
%Y A356490 Cf. A350955, A350956.
%K A356490 sign
%O A356490 0,2
%A A356490 _Stefano Spezia_, Aug 09 2022