A133156 Irregular triangle read by rows: coefficients of U(n,x), Chebyshev polynomials of the second kind with exponents in decreasing order.

1, 2, 4, -1, 8, -4, 16, -12, 1, 32, -32, 6, 64, -80, 24, -1, 128, -192, 80, -8, 256, -448, 240, -40, 1, 512, -1024, 672, -160, 10, 1024, -2304, 1792, -560, 60, -1, 2048, -5120, 4608, -1792, 280, -12, 4096, -11264, 11520, -5376, 1120, -84, 1

Offset

0,2

Comments

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).

From Gary W. Adamson, Nov 28 2008: (Start)

Triangle read by rows, unsigned = A000012 * A028297.

Row sums of absolute values give the Pell series, A000129.

(End)

The row sums are {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...}.

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

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

Links

Table of n, a(n) for n=0..48.

Tracale Austin, Hans Bantilan, Isao Jonas and Paul Kory, The Pfaffian Transformation, Journal of Integer Sequences, Vol. 12 (2009), page 25

P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014. - From Tom Copeland, Oct 11 2014

Pantelis A. Damianou, A Beautiful Sine Formula, Amer. Math. Monthly 121 (2014), no. 2, 120-135. MR3149030

Caglar Koca and Ozgur B. Akan, Modelling 1D Partially Absorbing Boundaries for Brownian Molecular Communication Channels, arXiv:2402.15888 [q-bio.MN], 2024. See p. 9.

Wikipedia, Chebyshev polynomials

Formula

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.

T(n,m) = (-1)^m*binomial(n - m, m)*2^(n - 2*m). - Roger L. Bagula and Gary W. Adamson, Dec 19 2008

From Tom Copeland, Feb 11 2016: (Start)

Shifted o.g.f.: G(x,t) = x/(1 - 2x + tx^2).

A053117 is a reflected, aerated version of this entry; A207538, an unsigned version; and A099089, a reflected, shifted version.

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)

Example

The first few Chebyshev polynomials of the second kind are
    1;
    2x;
    4x^2 -    1;
    8x^3 -    4x;
   16x^4 -   12x^2 +   1;
   32x^5 -   32x^3 +   6x;
   64x^6 -   80x^4 +  24x^2 -   1;
  128x^7 -  192x^5 +  80x^3 -   8x;
  256x^8 -  448x^6 + 240x^4 -  40x^2 +  1;
  512x^9 - 1024x^7 + 672x^5 - 160x^3 + 10x;
  ...
From Roger L. Bagula and Gary W. Adamson: (Start)
     1;
     2;
     4,    -1;
     8,    -4;
    16,   -12,    1;
    32,   -32,    6;
    64,   -80,   24,   -1;
   128,  -192,   80,   -8;
   256,  -448,  240,  -40,  1;
   512, -1024,  672, -160, 10;
  1024, -2304, 1792, -560, 60, -1; (End)
From  Philippe Deléham, Dec 27 2011: (Start)
Triangle (2, 0, 0, 0, 0, ...) DELTA (0, -1/2, 1/2, 0, 0, 0, 0, 0, ...) begins:
   1;
   2,   0;
   4,  -1,  0;
   8,  -4,  0,  0;
  16, -12,  1,  0,  0;
  32, -32,  6,  0,  0,  0;
  64, -80, 24, -1,  0,  0,  0; (End)

Mathematica

t[n_, m_] = (-1)^m*Binomial[n - m, m]*2^(n - 2*m);
Table[Table[t[n, m], {m, 0, Floor[n/2]}], {n, 0, 10}];
Flatten[%] (* Roger L. Bagula, Dec 19 2008 *)

Crossrefs

Cf. A038207, A053117.

Cf. A018297, A000129. - Gary W. Adamson, Nov 28 2008

Cf. A000079, A001787, A001788, A001789, A003472, A054849, A002409, A054851. - Philippe Deléham, Sep 12 2009

Cf. A091894, A097610, A099089, A207538.

Sequence in context: A125810 A226504 A152195 * A207538 A348869 A127529

Adjacent sequences: A133153 A133154 A133155 * A133157 A133158 A133159

Keyword

tabf,sign

Author

Gary W. Adamson, Dec 16 2007

Extensions

More terms from Philippe Deléham, Sep 12 2009

Status

approved