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
Examples
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
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.
- Index entries for sequences related to Chebyshev polynomials.
Programs
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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
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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]]]
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
Comments