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A334432
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Irregular triangle read by rows: T(m,k) gives the coefficients of x^k of the minimal polynomials of (2*cos(Pi/(2*m+1)))^2 = rho(2*n+1)^2, for m >= 0.
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2
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-4, 1, -1, 1, 1, -3, 1, -1, 6, -5, 1, -1, 9, -6, 1, -1, 15, -35, 28, -9, 1, 1, -21, 70, -84, 45, -11, 1, 1, -24, 26, -9, 1, 1, -36, 210, -462, 495, -286, 91, -15, 1, -1, 45, -330, 924, -1287, 1001, -455, 120, -17, 1, 1, -48, 148, -146, 64, -13, 1
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OFFSET
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0,1
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COMMENTS
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The length of row m is delta(2*m+1) + 1 = A055034(2*m+1) + 1.
For details see A334429, where the formula for the minimal polynomial MPc2(m, x) of 2*cos(Pi/(2*m+1))^2 = rho(2*m+1)^2, for m >= 0, is given.
The companion triangle for even n is A334431.
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LINKS
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FORMULA
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T(m, k) = [x^k] MPc2odd(m, x), with MPc2odd(m, x) = Product_{j=1..delta(2*m+1)} (x - (2 + R(rpnodd(2*m+1)_j, rho(2*m+1)))) (evaluated using C(2*m+1, rho(2*m+1)) = 0), for m >= 1, and MPc2odd(0, x) = -4 + x. Here R(n, x) is the monic Chebyshev R polynomial with coefficients given in A127672. C(n, x) is the minimal polynomial of rho(n) = 2*cos(Pi/n) given in A187360, and rpnodd(m) is the list of positive odd numbers coprime to 2*m + 1 and <= 2*m - 1.
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EXAMPLE
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The irregular triangle T(m,k) begins:
m, n \ k 0 1 2 3 4 5 6 7 8 9 ...
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0, 1 -4 1
1, 3: -1 1
2, 5: 1 -3 1
3, 7: -1 6 -5 1
4, 9: -1 9 -6 1
5, 11: -1 15 -35 28 -9 1
6, 13: 1 -21 70 -84 45 -11 1
7, 15: 1 -24 26 -9 1
8, 17: 1 -36 210 -462 495 -286 91 -15 1
9, 19: -1 45 -330 924 -1287 1001 -455 120 -17 1
10, 21: 1 -48 148 -146 64 -13 1
...
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CROSSREFS
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KEYWORD
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sign,tabf,easy
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AUTHOR
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STATUS
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approved
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