A327923 Irregular triangle read by rows: Coefficients of Schick's polynomials P(n, y^2), for n >= 1.

1, 1, -2, -1, 10, -24, 16, 1, -42, 504, -2640, 7040, -9984, 7168, -2048, -1, 170, -8568, 201552, -2687360, 22573824, -127339520, 502081536, -1417641984, 2901606400, -4310958080, 4600627200, -3435134976, 1702887424, -503316480, 67108864

Offset: 1

Author

Wolfdieter Lang, Nov 19 2019

Keywords

• sign
• easy
• tabf

Comments

The length of row n of this irregular triangle t(n, k) is 2^(n-1) = A000079(n-1), n >= 1.

These polynomials P(n, y^2) = Sum_{k=0..2^(n-1)-1} t(n, k)*y^2 appear in table 2 (Tabelle 2), p. 157, of Carl Schick's book as x_n/(2^n*x*y), for n >= 1, and x_0 = x. The polynomials y_n of Tabelle 1 on p. 156 appear as y_n = -Sum_{n=0..2^(n-1)} A321369(n, k)*y^(2*k), for n >= 1, with y_0 = y, and are related to the n-th iteration of the Chebyshev polynomial -T(2, y) = 1 - 2*x^2.

Originally the polynomials y_n and x_n are defined in Schick's rare notebook as: y_n = 1 - 2*(y_{n-1})^2, for n >= 1, with y_0 = y, and x_n/(2^n*x*y) = Product_{j=1..n-1} y_j, for n >= 1, and x_0 = x. The (x_n, y_n) come from coordinates of points on the unit circle with x_n = cos(phi_n) and y_n = sin(phi_n), with some initial condition (x_0, y_0) = (x, y).

Examples

			The irregular triangle T(n, k) begins:
n\k   0    1    2      3     4      5     6      7 ...
------------------------------------------------------
1:    1
2:    1   -2
3:   -1   10  -24     16
4:    1  -42  504  -2640  7040  -9984  7168  -2048
...
Row n = 5: -1  170  -8568  201552  -2687360  22573824  -127339520 502081536  -1417641984  2901606400  -4310958080 4600627200  -3435134976  1702887424  -503316480  67108864.
...
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References

• Carl Schick, Trigonometrie und unterhaltsame Zahlentheorie, Bokos Druck, Zürich, 2003 (ISBN 3-9522917-0-6).

Crossrefs

Cf. A000079, A053120, A321369.

Formula

Row polynomials: P(n, y^2) = Product_{j=1..n-1} y_j(y^2), for n >= 1 (the empty product equals 1), where y_j(y^2) = -T^{[j]}(2, y) = -T(2^j, y), the j-th iteration of -T(2, y) = - 1 + 2*y^2, with Chebyshev's T polynomials (A053120).

Row polynomials linearized in T: P(n, y^2) = (-1)^(n-1)*(1/2^(n-2)) * Sum_{m=1..2^(n-2)} T(2*(2^(n-1) + 1 - 2*m), y), for n >= 2, and P(1, y^2) = 1. See the irregular triangle A261693(n-1, m) = 2^(n-1) + 1 - 2*m, n >= 2, 1 <= m <= 2^(n-2).

Irregular triangle: t(n, k) = [y^(2*k)] P(n, y^2), n >= 1, k = 0, 1, ..., 2^(n-1)-1.