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%I #3 Dec 26 2012 22:15:09
%S 1,1,-1,1,-2,-3,1,-3,-15,-18,1,-4,-44,-144,-160,1,-5,-100,-625,-1750,
%T -1875,1,-6,-195,-1980,-10044,-25920,-27216,1,-7,-343,-5145,-40817,
%U -184877,-453789,-470596,1,-8,-560,-11648,-132608,-917504,-3866624,-9175040,-9437184,1,-9,-864,-23814,-367416,-3582306
%N Triangular table containing values of coefficients of the characteristic polynomial of a certain n x n circulant matrix, read by rows.
%C This is a lower triangular table.
%D Daniel Zwillinger, ed., "CRC Standard Mathematical Tables and Formulae", 31st Edition, ISBN 1-58488-291, Section 2.6.2.25 (page 141) and Section 2.6.11.3 (page 152).
%F First column is unity. Second column (A127407) is a(n+1) = n*(n+1)^2*(n+8)/(2*3!) for n>=1. Third column (A127408) is a(n+2) = n*(n+1)*(n+2)^3*(2n+14)/(2 * 4!) for n>=1. In general, k-th column is given by a(n+(k-1)) = n*(n+1)*(n+2)*...*(n+(k-1))^k*((k-1)n+S(k))/(2 * (k+1)!) for n>=1, where S(k) is the k-th term of A014206.
%e The third row represents the coefficients of the characteristic polynomial of [1 2 3; 3 1 2; 2 3 1], which is x^3 - 3*x^2 - 15*x - 18. Thus the row reads 1,-3,-15,-18.
%o (OCTAVE, MATLAB) for n:0:N a = round(poly(gallery('circul',1:n))); end (OCTAVE, MATLAB) n * (n+1)^2 * (n+8) / (2 * factorial(3)); n * (n+1) * (n+2)^3 * (2*n + 14) / (2 * factorial(4)); n * (n+1) * (n+2) * (n+3)^4 * (3*n + 22) / (2 * factorial(5)); n * (n+1) * (n+2) * (n+3) * (n+4)^5 * (4*n + 32) / (2 * factorial(6));
%Y Cf. A000142, A014206, A127407, A127408, A127409, A127410, A127411.
%K easy,sign,tabl
%O 0,5
%A _Paul Max Payton_, Feb 09 2007
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