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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).
2
0, -1, 28, -1360, 105500, -12051585, 1908871832, -400855203840, 107838796034520, -36175347978515625, 14806446317943766420, -7263073394295238840320, 4206546078973080241293076, -2840250692354398785860048105, 2211476237421629752792968750000
OFFSET
1,3
LINKS
Eric Weisstein's World of Mathematics, Circulant Matrix.
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]
EXAMPLE
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|.
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 *)
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.
Sequence in context: A281328 A091993 A213692 * A249348 A366302 A013926
KEYWORD
easy,sign
AUTHOR
Jonathan Vos Post, May 20 2006
EXTENSIONS
More terms from Alois P. Heinz, Mar 16 2017
STATUS
approved