A334432 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.

-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

Offset: 0

Author

Wolfdieter Lang, Jun 15 2020

Keywords

• sign
• tabf
• easy

Comments

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.

Examples

			The irregular triangle T(m,k) begins:
m,   n \ k  0    1    2     3     4     5     6   7   8  9 ...
--------------------------------------------------------------
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
...
		

Crossrefs

Cf. A055034, A334429, A334431.

Formula

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.