%I #20 Jul 29 2023 06:35:32
%S 1,1,3,0,-256,-5000,-46656,941192,67108864,2066242608,24000000000,
%T -1659995174464,-142657607172096,-5964309791355136,-76196618232397824,
%U 11210083593750000000,1180591620717411303424,62286325600853591655680,839390038939659468275712,-213252813410122222659258368
%N a(n) is the determinant of an n X n Hermitian Toeplitz matrix whose first row consists of n, (n-1)*i, (n-2)*i, ..., 3*i, 2*i, i, where i denotes the imaginary unit.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>
%F A359616(n) <= a(n) <= A359617(n).
%e a(3) = 0:
%e [ 3, 2*i, i;
%e -2*i, 3, 2*i;
%e -i, -2*i, 3 ]
%p A359561 := proc(n)
%p local T,c,r ;
%p if n =0 then
%p return 1 ;
%p end if;
%p T := Matrix(n,n) ;
%p T[1,1] := n ;
%p for c from 2 to n do
%p T[1,c] := (n-c+1)*I ;
%p end do:
%p for r from 2 to n do
%p for c from 1 to r-1 do
%p T[r,c] := -T[c,r] ;
%p end do:
%p for c from r to n do
%p T[r,c] := T[r-1,c-1] ;
%p end do:
%p end do:
%p LinearAlgebra[Determinant](T) ;
%p simplify(%) ;
%p end proc:
%p seq(A359561(n),n=0..25) ; # _R. J. Mathar_, Jan 31 2023
%t Join[{1},Table[Det[ToeplitzMatrix[Join[{n},I Reverse[Range[n-1]]]]],{n,19}]]
%o (Python)
%o from sympy import Matrix, I
%o def A359561(n): return Matrix(n,n,[(n+j-i if i>j else j-i-n) if i!=j else n*I for i in range(n) for j in range(n)]).det()*(1,-I,-1,I)[n&3] # _Chai Wah Wu_, Jan 25 2023
%Y Cf. A307783 (symmetric Toeplitz matrix).
%Y Cf. A359559, A359560, A359562 (permanent).
%Y Cf. A359616 (minimal), A359617 (maximal).
%K sign
%O 0,3
%A _Stefano Spezia_, Jan 06 2023