A334429 -id:A334429 - cp's OEIS frontend

A334431 Irregular triangle read by rows: T(m,k) gives the coefficients of x^k of the minimal polynomials of (2cos(Pi/(2m)))^2, for m >= 1.

0, 1, -2, 1, -3, 1, 2, -4, 1, 5, -5, 1, 1, -4, 1, -7, 14, -7, 1, 2, -16, 20, -8, 1, -3, 9, -6, 1, 1, -12, 19, -8, 1, -11, 55, -77, 44, -11, 1, 1, -16, 20, -8, 1

Offset: 1

Wolfdieter Lang, Jun 15 2020

The length of row m is delta(m) + 1 = A055034(m) + 1.
For details see A334429, where the formula for the minimal polynomial MPc2(m, x) of 2*cos(Pi/(2*m))^2 = rho(2*m)^2 is given.
The companion triangle for odd n is A334432.

			The irregular triangle T(m, k) begins:
m,   n \ k  0   1   2    3   4    5   6 ...
-------------------------------------------
1,   2:     0   1
2,   4:    -2   1
3,   6:    -3   1
4,   8:     2  -4   1
5,  10:     5  -5   1
6,  12:     1  -4   1
7,  14:    -7  14  -7    1
8,  16:     2 -16  20   -8   1
9,  18:    -3   9  -6    1
10, 20:     1 -12  19   -8   1
11, 22:   -11  55 -77   44 -11    1
12, 24:     1 -16  20   -8   1
13, 26:    13 -91 182 -156  65  -13   1
14, 28:     1 -24  86 -104  53  -12   1
15, 30:     1  -8  14   -7   1
...

Cf. A055034, A334429, A334432.

T(m, k) = [x^k] MPc2even(m, x), with MPc2even(m, x) = Product_{j=1..delta(m)} (x - (2 + R(rpnodd(m)_j, rho(m)))) (evaluated using C(m, rho(m)) = 0), for m >= 2, and MPc2even(1, x) = 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 m and <= m - 1.

A334432 Irregular triangle read by rows: T(m,k) gives the coefficients of x^k of the minimal polynomials of (2cos(Pi/(2m+1)))^2 = rho(2*n+1)^2, for m >= 0.

Original entry on oeis.org

-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

Wolfdieter Lang, Jun 15 2020

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.

			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
...

Cf. A055034, A334429, A334431.

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.

cp's OEIS Frontend

A334431 Irregular triangle read by rows: T(m,k) gives the coefficients of x^k of the minimal polynomials of (2cos(Pi/(2m)))^2, for m >= 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A334432 Irregular triangle read by rows: T(m,k) gives the coefficients of x^k of the minimal polynomials of (2cos(Pi/(2m+1)))^2 = rho(2*n+1)^2, for m >= 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

cp's OEIS Frontend

A334431 Irregular triangle read by rows: T(m,k) gives the coefficients of x^k of the minimal polynomials of (2*cos(Pi/(2*m)))^2, for m >= 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A334431 Irregular triangle read by rows: T(m,k) gives the coefficients of x^k of the minimal polynomials of (2cos(Pi/(2m)))^2, for m >= 1.

A334432 Irregular triangle read by rows: T(m,k) gives the coefficients of x^k of the minimal polynomials of (2cos(Pi/(2m+1)))^2 = rho(2*n+1)^2, for m >= 0.