cp's OEIS Frontend

A118709 a(n) = determinant of n X n circulant matrix whose first row is the first n cube numbers 0, 1, ..., (n-1)^3.

%I A118709 #10 Feb 16 2025 08:33:01
%S A118709 0,-1,513,-532800,1077540500,-3831689610000,22051842087895137,
%T A118709 -192710430555501494272,2433436736207275231050384,
%U A118709 -42684202683959414242500000000,1007311823853329619224620155226025,-31149342348518897782279760206406615040
%N A118709 a(n) = determinant of n X n circulant matrix whose first row is the first n cube numbers 0, 1, ..., (n-1)^3.
%H A118709 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CirculantMatrix.html">Circulant Matrix</a>.
%F A118709 Contribution from Missouri State University Problem-Solving Group (MSUPSG(AT)MissouriState.edu), May 05 2010: (Start)
%F A118709 a(n) = (-1)^(n-1)*(n-1)^2*n^(n-2)*(n^(2n)-b(n)^n-c(n)^n+(n^2-3n+3)^n)/24
%F A118709 where
%F A118709 b(n)=(2*n^2-3*n-3+sqrt(15n^2-18n-9)i)/2 and
%F A118709 c(n)=(2*n^2-3*n-3-sqrt(15n^2-18n-9)i)/2 (End)
%e A118709 a(2) = -1 because of the determinant -1 =
%e A118709 | 0, 1 |
%e A118709 | 1, 0 |.
%e A118709 a(3) = 513 = determinant
%e A118709 |0,1,8|
%e A118709 |8,0,1|
%e A118709 |1,8,0|.
%e A118709 a(6) = 22051842087895137 = determinant
%e A118709 |0,1,8,27,64,125,216|
%e A118709 |216,0,1,8,27,64,125|
%e A118709 |125,216,0,1,8,27,64|
%e A118709 |64,125,216,0,1,8,27|
%e A118709 |27,64,125,216,0,1,8|
%e A118709 |8,27,64,125,216,0,1|
%e A118709 |1,8,27,64,125,216,0|.
%t A118709 Table[Det[Table[RotateRight[Range[0,i]^3,n],{n,0,i}]],{i,0,10}] (* _Harvey P. Dale_, Oct 22 2012 *)
%Y A118709 See also: A000578 The cubes: a(n) = n^3. 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 A118709 Cf. A000578, A048954, A052182, A066933, A086459, A086569.
%K A118709 easy,sign
%O A118709 1,3
%A A118709 _Jonathan Vos Post_, May 20 2006
%E A118709 More terms from _Harvey P. Dale_, Oct 22 2012