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A359561 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.

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