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
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.
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
KEYWORD
sign,tabf,easy
AUTHOR
Doug Hensley, Nov 07 2018
STATUS
approved