A118705 a(n) = determinant of n X n circulant matrix whose first row is the first n triangular numbers A000217(0), A000217(1), ... A000217(n-1).

0, -1, 28, -1360, 105500, -12051585, 1908871832, -400855203840, 107838796034520, -36175347978515625, 14806446317943766420, -7263073394295238840320, 4206546078973080241293076, -2840250692354398785860048105, 2211476237421629752792968750000

Offset: 1

Author

Jonathan Vos Post, May 20 2006

Keywords

• easy
• sign

Examples

			a(2) = - 1 because of the determinant -1 =
  | 0, 1 |
  | 1, 0 |.
a(4) = -1360 = determinant
  |0,1,3,6|
  |6,0,1,3|
  |3,6,0,1|
  |1,3,6,0|.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..226
• Eric Weisstein's World of Mathematics, Circulant Matrix.

Crossrefs

See also: A048954 Wendt determinant of n-th circulant matrix C(n). A052182 Circulant of natural numbers. A066933 Circulant of prime numbers. A086459 Circulant of powers of 2.

Cf. A000045, A048954, A052182, A066933, A086459, A086569.

Programs

Maple

f:= proc(n) uses LinearAlgebra;local i;
  Determinant(Matrix(n, shape=Circulant[[seq(i*(i+1)/2, i=0..n-1)]]))
end proc:
map(f, [$1..30]); # Robert Israel, Jan 25 2023

Mathematica

r[n_] := r[n] = Table[k(k+1)/2, {k, 0, n-1}];
M[n_] := Table[RotateRight[r[n], m-1], {m, 1, n}];
a[n_] := Det[M[n]];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 11 2023 *)

Formula

a(n) = (-1)^(n-1)*n^(n-2)*(n+1)*(n-1)*((n+1)^n-(n-1)^n)/(6*2^n). [Missouri State University Problem-Solving Group (MSUPSG(AT)MissouriState.edu), May 03 2010]

Extensions

More terms from Alois P. Heinz, Mar 16 2017