A127412 Triangular table containing values of coefficients of the characteristic polynomial of a certain n x n circulant matrix, read by rows.

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

Offset

0,5

Comments

This is a lower triangular table.

References

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).

Links

Table of n, a(n) for n=0..50.

Formula

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.

Example

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.

Prog

(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));

Crossrefs

Cf. A000142, A014206, A127407, A127408, A127409, A127410, A127411.

Sequence in context: A126744 A126736 A253257 * A304791 A306646 A152832

Adjacent sequences: A127409 A127410 A127411 * A127413 A127414 A127415

Keyword

easy,sign,tabl

Author

Paul Max Payton, Feb 09 2007

Status

approved