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A122374
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Triangle in which row n gives the coefficients of det(A-xI), where A is the n X n matrix with 1's on antidiagonal and last row and column, 0's elsewhere.
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0
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1, 1, -1, -1, -1, 1, -1, 1, 2, -1, 1, -1, -4, -1, 1, 1, -3, -3, 4, 2, -1, -1, 3, 7, -2, -7, -1, 1, -1, 5, 4, -11, -5, 7, 2, -1, 1, -5, -10, 9, 18, -3, -10, -1, 1, 1, -7, -5, 22, 9, -24, -7, 10, 2, -1, -1, 7, 13, -20, -34, 18, 34, -4, -13, -1, 1, -1, 9, 6, -37, -14, 58, 16, -42, -9, 13, 2, -1, 1, -9, -16, 35, 55, -50, -80, 30, 55, -5, -16, -1, 1
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OFFSET
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0,9
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LINKS
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EXAMPLE
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Triangle starts:
{ 1},
{ 1, -1},
{-1, -1, 1},
{-1, 1, 2, -1},
{ 1, -1, -4, -1, 1},
{ 1, -3, -3, 4, 2, -1},
{-1, 3, 7, -2, -7, -1, 1},
{-1, 5, 4, -11, -5, 7, 2, -1},
{ 1, -5, -10, 9, 18, -3, -10, -1, 1},
{ 1, -7, -5, 22, 9, -24, -7, 10, 2, -1}
...
For n = 0, the determinant of the 0 X 0 matrix is 1 by convention, which yields row 0 = [ 1 ].
For n = 1, we have det [1 - x] = 1 - x, which yields row 1 = [1, -1].
For n = 2, we have det [-x, 1; 1, 1 - x] = x(x - 1) - 1 = x^2 - x - 1; in order of increasing powers this yields row 2 = [-1, -1, +1]. (End)
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PROG
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(PARI) A122374_row(n)=(-1)^n*Vecrev(charpoly(matrix(n, n, i, j, i==n||j==n||i+j==n+1), x)) \\ Yields the n-th row. - M. F. Hasler, Apr 26 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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