A321369 Coefficients of successive polynomials formed by iterating f(x) = -1 + 2x^2. Irregular triangle read by rows.

1, -1, 2, 1, -8, 8, 1, -32, 160, -256, 128, 1, -128, 2688, -21504, 84480, -180224, 212992, -131072, 32768, 1, -512, 43520, -1462272, 25798656, -275185664, 1926299648, -9313976320, 32133218304, -80648077312, 148562247680, -200655503360

Offset

0,3

Comments

For n >= 1 subsequence of Chebyshev T polynomials, the (2^n)-th ones, per Wikipedia article on Chebyshev polynomials, in particular, the nesting property. Relevant to Putnam problem B4 of 2000.

The length of row n is 1 for n = 0 and A000051(n-1) = 2^(n-1) + 1 for n >= 1.

This irregular triangle T(n, k) appears in Table 1 (Tabelle 1), p. 156, of the book by Carl Schick as polynomials y_n(y) = -Sum_{n=0..2^(n-1)} T(n, k) * y^(2*k), for n >= 1. There y_0 = y. - Wolfdieter Lang, Nov 15 2019

References

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

Links

Table of n, a(n) for n=0..31.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, electronic version, p. 795.

Wikipedia, Chebyshev polynomials.

Index entries for sequences related to Chebyshev polynomials.

Formula

T(0,0) = 1, T(n, k) = [x^(2*k)] T(2^n, x), with Chebyshev's T-polynomials (A053120). - Wolfdieter Lang, Oct 25 2019

Example

Polynomials are 1, then -1+2x^2, then -1+2(-1+2x^2)^2 = 1-8x^2+8x^4, etc. leading to array T:
n\k  0    1    2      3     4       5      6       7     8 ...
--------------------------------------------------------------
0:   1
1:  -1   2
2:   1  -8   8
3:   1  -32  160   -256   128
4:   1 -128 2688 -21504 84480 -180224 212992 -131072 32768
...
-------------------------------------------------------------------------
row n=5: 1 -512 43520 -1462272 25798656 -275185664 1926299648 -9313976320 32133218304 -80648077312 148562247680 -200655503360 196293427200 -135291469824 62277025792 -17179869184 2147483648. Reformatted and extended by Wolfdieter Lang, Oct 25 2019

Maple

P := proc(n) local t; if n = 0 then return 1 fi;
t := x -> orthopoly[T](2^n, x): seq(coeff(t(x), x, 2*k), k=0..2^(n-1)) end:
seq(P(n), n=0..5); # Peter Luschny, Oct 26 2019

Mathematica

h[x_] := 2 x^2 - 1;
a[n_, k_] :=
If[k == 0, 1, Coefficient[Expand[Nest[h, x, n]], x^(2 k)]];
b[n_] := Table[a[n, k], {k, 0, 2^(n - 1)}];
c[1] = {-1, 2}; c[n_] := c[n] = Join[c[n - 1], b[n]];
sequence[n_] :=
Module[{p, q, r}, p = 2; While[Length[c[p]] < n, p++]; c[p][[n]]]

Crossrefs

Cf. A000051, A053120, A075733, A127674.

Sequence in context: A127674 A271316 A145901 * A286724 A123516 A193604

Adjacent sequences: A321366 A321367 A321368 * A321370 A321371 A321372

Keyword

sign,tabf,easy

Author

Doug Hensley, Nov 07 2018

Status

approved