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

%I A321369 #34 Nov 17 2019 08:39:17
%S A321369 1,-1,2,1,-8,8,1,-32,160,-256,128,1,-128,2688,-21504,84480,-180224,
%T A321369 212992,-131072,32768,1,-512,43520,-1462272,25798656,-275185664,
%U A321369 1926299648,-9313976320,32133218304,-80648077312,148562247680,-200655503360
%N A321369 Coefficients of successive polynomials formed by iterating f(x) = -1 + 2x^2. Irregular triangle read by rows.
%C A321369 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.
%C A321369 The length of row n is 1 for n = 0 and A000051(n-1) = 2^(n-1) + 1 for n >= 1.
%C A321369 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
%D A321369 Carl Schick, Trigonometrie und unterhaltsame Zahlentheorie, Bokos Druck, Zürich, 2003 (ISBN 3-9522917-0-6).
%H A321369 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&amp;Page=795">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, electronic version, p. 795.
%H A321369 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.
%H A321369 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A321369 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
%e A321369 Polynomials are 1, then -1+2x^2, then -1+2(-1+2x^2)^2 = 1-8x^2+8x^4, etc. leading to array T:
%e A321369 n\k  0    1    2      3     4       5      6       7     8 ...
%e A321369 --------------------------------------------------------------
%e A321369 0:   1
%e A321369 1:  -1   2
%e A321369 2:   1  -8   8
%e A321369 3:   1  -32  160   -256   128
%e A321369 4:   1 -128 2688 -21504 84480 -180224 212992 -131072 32768
%e A321369 ...
%e A321369 -------------------------------------------------------------------------
%e A321369 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
%p A321369 P := proc(n) local t; if n = 0 then return 1 fi;
%p A321369 t := x -> orthopoly[T](2^n, x): seq(coeff(t(x), x, 2*k), k=0..2^(n-1)) end:
%p A321369 seq(P(n), n=0..5); # _Peter Luschny_, Oct 26 2019
%t A321369 h[x_] := 2 x^2 - 1;
%t A321369 a[n_, k_] :=
%t A321369 If[k == 0, 1, Coefficient[Expand[Nest[h, x, n]], x^(2 k)]];
%t A321369 b[n_] := Table[a[n, k], {k, 0, 2^(n - 1)}];
%t A321369 c[1] = {-1, 2}; c[n_] := c[n] = Join[c[n - 1], b[n]];
%t A321369 sequence[n_] :=
%t A321369 Module[{p, q, r}, p = 2; While[Length[c[p]] < n, p++]; c[p][[n]]]
%Y A321369 Cf. A000051, A053120, A075733, A127674.
%K A321369 sign,tabf,easy
%O A321369 0,3
%A A321369 _Doug Hensley_, Nov 07 2018