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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).

%I A118705 #17 Feb 16 2025 08:33:01
%S A118705 0,-1,28,-1360,105500,-12051585,1908871832,-400855203840,
%T A118705 107838796034520,-36175347978515625,14806446317943766420,
%U A118705 -7263073394295238840320,4206546078973080241293076,-2840250692354398785860048105,2211476237421629752792968750000
%N 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).
%H A118705 Robert Israel, <a href="/A118705/b118705.txt">Table of n, a(n) for n = 1..226</a>
%H A118705 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CirculantMatrix.html">Circulant Matrix</a>.
%F A118705 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]
%e A118705 a(2) = - 1 because of the determinant -1 =
%e A118705   | 0, 1 |
%e A118705   | 1, 0 |.
%e A118705 a(4) = -1360 = determinant
%e A118705   |0,1,3,6|
%e A118705   |6,0,1,3|
%e A118705   |3,6,0,1|
%e A118705   |1,3,6,0|.
%p A118705 f:= proc(n) uses LinearAlgebra;local i;
%p A118705   Determinant(Matrix(n, shape=Circulant[[seq(i*(i+1)/2, i=0..n-1)]]))
%p A118705 end proc:
%p A118705 map(f, [$1..30]); # _Robert Israel_, Jan 25 2023
%t A118705 r[n_] := r[n] = Table[k(k+1)/2, {k, 0, n-1}];
%t A118705 M[n_] := Table[RotateRight[r[n], m-1], {m, 1, n}];
%t A118705 a[n_] := Det[M[n]];
%t A118705 Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Feb 11 2023 *)
%Y A118705 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.
%Y A118705 Cf. A000045, A048954, A052182, A066933, A086459, A086569.
%K A118705 easy,sign
%O A118705 1,3
%A A118705 _Jonathan Vos Post_, May 20 2006
%E A118705 More terms from _Alois P. Heinz_, Mar 16 2017