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A127412
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Triangular table containing values of coefficients of the characteristic polynomial of a certain n x n circulant matrix, read by rows.
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5
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1, 1, -1, 1, -2, -3, 1, -3, -15, -18, 1, -4, -44, -144, -160, 1, -5, -100, -625, -1750, -1875, 1, -6, -195, -1980, -10044, -25920, -27216, 1, -7, -343, -5145, -40817, -184877, -453789, -470596, 1, -8, -560, -11648, -132608, -917504, -3866624, -9175040, -9437184, 1, -9, -864, -23814, -367416, -3582306
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OFFSET
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0,5
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COMMENTS
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This is a lower triangular table.
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REFERENCES
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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).
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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PROG
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(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));
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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