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%I #65 Mar 01 2024 01:54:57
%S 1,2,4,-1,8,-4,16,-12,1,32,-32,6,64,-80,24,-1,128,-192,80,-8,256,-448,
%T 240,-40,1,512,-1024,672,-160,10,1024,-2304,1792,-560,60,-1,2048,
%U -5120,4608,-1792,280,-12,4096,-11264,11520,-5376,1120,-84,1
%N Irregular triangle read by rows: coefficients of U(n,x), Chebyshev polynomials of the second kind with exponents in decreasing order.
%C The Chebyshev polynomials of the second kind are defined by the recurrence relation: U(0,x) = 1; U(1,x) = 2x; U(n+1,x) = 2x*U(n,x) - U(n-1,x).
%C From _Gary W. Adamson_, Nov 28 2008: (Start)
%C Triangle read by rows, unsigned = A000012 * A028297.
%C Row sums of absolute values give the Pell series, A000129.
%C (End)
%C The row sums are {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...}.
%C Triangle, with zeros omitted, given by (2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Dec 27 2011
%C Coefficients in the expansion of sin((n+1)*x)/sin(x) in descending powers of cos(x). The length of the n-th row is A008619(n). - _Jianing Song_, Nov 02 2018
%H Tracale Austin, Hans Bantilan, Isao Jonas and Paul Kory, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Egge/egge8.html">The Pfaffian Transformation</a>, Journal of Integer Sequences, Vol. 12 (2009), page 25
%H P. Damianou, <a href="http://arxiv.org/abs/1110.6620">On the characteristic polynomials of Cartan matrices and Chebyshev polynomials</a>, arXiv preprint arXiv:1110.6620 [math.RT], 2014. - From _Tom Copeland_, Oct 11 2014
%H Pantelis A. Damianou, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.121.02.120">A Beautiful Sine Formula</a>, Amer. Math. Monthly 121 (2014), no. 2, 120-135. MR3149030
%H Caglar Koca and Ozgur B. Akan, <a href="https://arxiv.org/abs/2402.15888">Modelling 1D Partially Absorbing Boundaries for Brownian Molecular Communication Channels</a>, arXiv:2402.15888 [q-bio.MN], 2024. See p. 9.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>
%F A generating function for U(n) is 1/(1 - 2tx + t^2). Given A038207, shift down columns to allow for (1, 1, 2, 2, 3, 3, ...) terms in each row, then insert alternate signs.
%F T(n,m) = (-1)^m*binomial(n - m, m)*2^(n - 2*m). - _Roger L. Bagula_ and _Gary W. Adamson_, Dec 19 2008
%F From _Tom Copeland_, Feb 11 2016: (Start)
%F Shifted o.g.f.: G(x,t) = x/(1 - 2x + tx^2).
%F A053117 is a reflected, aerated version of this entry; A207538, an unsigned version; and A099089, a reflected, shifted version.
%F The compositional inverse of G(x,t) is Ginv(x,t) = ((1 + 2x) - sqrt((1 + 2x)^2 - 4tx^2))/(2tx) = x - 2x^2 + (4 + t)x^3 - (8 + 6t)x^4 + ..., a shifted o.g.f. for A091894 (mod signs with A091894(0,0) = 0.). Cf. A097610 with h_1 = -2 and h_2 = t. (End)
%e The first few Chebyshev polynomials of the second kind are
%e 1;
%e 2x;
%e 4x^2 - 1;
%e 8x^3 - 4x;
%e 16x^4 - 12x^2 + 1;
%e 32x^5 - 32x^3 + 6x;
%e 64x^6 - 80x^4 + 24x^2 - 1;
%e 128x^7 - 192x^5 + 80x^3 - 8x;
%e 256x^8 - 448x^6 + 240x^4 - 40x^2 + 1;
%e 512x^9 - 1024x^7 + 672x^5 - 160x^3 + 10x;
%e ...
%e From _Roger L. Bagula_ and _Gary W. Adamson_: (Start)
%e 1;
%e 2;
%e 4, -1;
%e 8, -4;
%e 16, -12, 1;
%e 32, -32, 6;
%e 64, -80, 24, -1;
%e 128, -192, 80, -8;
%e 256, -448, 240, -40, 1;
%e 512, -1024, 672, -160, 10;
%e 1024, -2304, 1792, -560, 60, -1; (End)
%e From _Philippe Deléham_, Dec 27 2011: (Start)
%e Triangle (2, 0, 0, 0, 0, ...) DELTA (0, -1/2, 1/2, 0, 0, 0, 0, 0, ...) begins:
%e 1;
%e 2, 0;
%e 4, -1, 0;
%e 8, -4, 0, 0;
%e 16, -12, 1, 0, 0;
%e 32, -32, 6, 0, 0, 0;
%e 64, -80, 24, -1, 0, 0, 0; (End)
%t t[n_, m_] = (-1)^m*Binomial[n - m, m]*2^(n - 2*m);
%t Table[Table[t[n, m], {m, 0, Floor[n/2]}], {n, 0, 10}];
%t Flatten[%] (* _Roger L. Bagula_, Dec 19 2008 *)
%Y Cf. A038207, A053117.
%Y Cf. A018297, A000129. - _Gary W. Adamson_, Nov 28 2008
%Y Cf. A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851. - _Philippe Deléham_, Sep 12 2009
%Y Cf. A091894, A097610, A099089, A207538.
%K tabf,sign
%O 0,2
%A _Gary W. Adamson_, Dec 16 2007
%E More terms from _Philippe Deléham_, Sep 12 2009
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