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%I #16 Feb 11 2023 09:24:41
%S 0,-1,28,-1360,105500,-12051585,1908871832,-400855203840,
%T 107838796034520,-36175347978515625,14806446317943766420,
%U -7263073394295238840320,4206546078973080241293076,-2840250692354398785860048105,2211476237421629752792968750000
%N 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 Robert Israel, <a href="/A118705/b118705.txt">Table of n, a(n) for n = 1..226</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CirculantMatrix.html">Circulant Matrix</a>.
%F 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 a(2) = - 1 because of the determinant -1 =
%e | 0, 1 |
%e | 1, 0 |.
%e a(4) = -1360 = determinant
%e |0,1,3,6|
%e |6,0,1,3|
%e |3,6,0,1|
%e |1,3,6,0|.
%p f:= proc(n) uses LinearAlgebra;local i;
%p Determinant(Matrix(n, shape=Circulant[[seq(i*(i+1)/2, i=0..n-1)]]))
%p end proc:
%p map(f, [$1..30]); # _Robert Israel_, Jan 25 2023
%t r[n_] := r[n] = Table[k(k+1)/2, {k, 0, n-1}];
%t M[n_] := Table[RotateRight[r[n], m-1], {m, 1, n}];
%t a[n_] := Det[M[n]];
%t Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Feb 11 2023 *)
%Y 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 Cf. A000045, A048954, A052182, A066933, A086459, A086569.
%K easy,sign
%O 1,3
%A _Jonathan Vos Post_, May 20 2006
%E More terms from _Alois P. Heinz_, Mar 16 2017
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