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A359560 a(n) is the permanent of an n X n Hermitian Toeplitz matrix whose first row consists of 1, 2*i, ..., n*i, where i denotes the imaginary unit.

%I A359560 #23 Jan 31 2023 05:13:21
%S A359560 1,1,5,18,360,2800,151424,1926704,218991568,3961998320,815094714320,
%T A359560 19339258670304,6524060415099520,192715406460607360,
%U A359560 99364368150722162944,3525158026102570745600,2635328330670632415828224,109381927750670379873854720,113797518402277434839782802688
%N A359560 a(n) is the permanent of an n X n Hermitian Toeplitz matrix whose first row consists of 1, 2*i, ..., n*i, where i denotes the imaginary unit.
%H A359560 Wikipedia, <a href="https://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>
%F A359560 A359614(n) <= a(n) <= A359615(n).
%e A359560 a(3) = 18:
%e A359560   [   1,  2*i, 3*i;
%e A359560    -2*i,    1, 2*i;
%e A359560    -3*i, -2*i,   1 ]
%p A359560 A359560 := proc(n)
%p A359560     local T,c,r ;
%p A359560     if n =0 then
%p A359560         return 1 ;
%p A359560     end if;
%p A359560     T := Matrix(n,n,shape=hermitian) ;
%p A359560     T[1,1] := 1 ;
%p A359560     for c from 2 to n do
%p A359560         T[1,c] := c*I ;
%p A359560     end do:
%p A359560     for r from 2 to n do
%p A359560         for c from r to n do
%p A359560             T[r,c] := T[r-1,c-1] ;
%p A359560         end do:
%p A359560     end do:
%p A359560     LinearAlgebra[Permanent](T) ;
%p A359560     simplify(%) ;
%p A359560 end proc:
%p A359560 seq(A359560(n),n=0..15) ; # _R. J. Mathar_, Jan 31 2023
%t A359560 Join[{1},Table[Permanent[ToeplitzMatrix[Join[{1},I Range[2,n]]]],{n,18}]]
%o A359560 (PARI) a(n) = matpermanent(matrix(n, n, i, j, if (i==j, 1, if (i<j, I*(j-i+1), I*(j-i-1))))); \\ _Michel Marcus_, Jan 20 2023
%o A359560 (Python)
%o A359560 from sympy import Matrix, I
%o A359560 def A359560(n): return Matrix(n,n,[i-j+(1 if i>j else -1) if i!=j else I for i in range(n) for j in range(n)]).per()*(1,-I,-1,I)[n&3] if n else 1 # _Chai Wah Wu_, Jan 25 2023
%Y A359560 Cf. A143182, A204235 (symmetric Toeplitz matrix).
%Y A359560 Cf. A359559 (determinant), A359561, A359562.
%Y A359560 Cf. A359614 (minimal), A359615 (maximal).
%K A359560 nonn
%O A359560 0,3
%A A359560 _Stefano Spezia_, Jan 06 2023