A053118 -id:A053118 - cp's OEIS frontend

A053117 Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).

1, 0, 2, -1, 0, 4, 0, -4, 0, 8, 1, 0, -12, 0, 16, 0, 6, 0, -32, 0, 32, -1, 0, 24, 0, -80, 0, 64, 0, -8, 0, 80, 0, -192, 0, 128, 1, 0, -40, 0, 240, 0, -448, 0, 256, 0, 10, 0, -160, 0, 672, 0, -1024, 0, 512, -1, 0, 60, 0, -560, 0, 1792, 0, -2304, 0, 1024, 0, -12, 0, 280, 0, -1792, 0, 4608, 0, -5120, 0, 2048, 1, 0, -84, 0, 1120, 0, -5376, 0, 11520, 0, -11264, 0, 4096

Offset: 0

Wolfdieter Lang

G.f. for row polynomials U(n,x) (signed triangle): 1/(1-2*x*z+z^2). Unsigned triangle |a(n,m)| has Fibonacci polynomials F(n+1,2*x) as row polynomials with g.f. 1/(1-2*x*z-z^2).
Row sums (unsigned triangle) A000129(n+1) (Pell). Row sums (signed triangle) A000027(n+1) (natural numbers).
The o.g.f. for the Legendre polynomials L(n,x) is 1 / sqrt(1- 2x*z + z^2), and squaring it gives the o.g.f. of this entry, so Sum_{k=0..n} L(k,x) L(n-k,x) = U(n,x). This reduces to U(n,x) = L(n/2,x)^2 + 2*Sum_{k=0...n/2-1} L(k,x) L(n-k,x) for n even and U(n,x) = 2*Sum_{k=0..(n-1)/2} L(k,x) L(n-k.x) for odd n. (Cf. also Allouche et al.) For a connection through the Legendre polynomials to elliptic curves and modular forms, see the MathOverflow question below. For the normalized Legendre polynomials, see A100258. (Cf. A097610 with h1 = -2x and h2 = 1, A207538, A099089 and A133156.) - Tom Copeland, Feb 04 2016
The compositional inverse of the shifted o.g.f. x / (1 + 2xz + z^2) for differently signed row polynomials of this entry is the shifted o.g.f. of A121448. The unsigned, non-vanishing antidiagonals (top to bottom) of this triangle are the rows of A038207. - Tom Copeland, Feb 08 2016

			Triangle begins:
   1;
   0,  2;
  -1,  0,   4;
   0, -4,   0, 8;
   1,  0, -12, 0, 16;
  ...
E.g., fourth row (n=3) {0,-4,0,8} corresponds to polynomial U(3,x) = -4*x + 8*x^3.

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 22, page 196.

T. D. Noe, Rows n=0..100 of triangle, flattened
J.-P. Allouche and G. Skordev, Schur congruences, Carlitz sequences of polynomials and automaticity, Discrete Mathematics, Vol. 214, Issue 1-3, 21 March 2000, pp. 21-49.
Paul Barry and A. Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, J. Int. Seq. 13 (2010) # 10.9.4, section 5.
P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014 (p. 10). - From _Tom Copeland_, Oct 11 2014
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
MathOverflow, Geometric picture of invariant differential of an elliptic curve, Dec 4 2011.
Valentin Ovsienko, Towards quantized complex numbers: q-deformed Gaussian integers and the Picard group, arXiv:2103.10800 [math.QA], 2021.
R. Pemantle and M. C. Wilson, Asymptotics of multivariate sequences, I: smooth points of the singular variety, arXiv:math/0003192 [math.CO], 2000.
A. Sapounakis, I. Tasoulas, and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
Index entries for sequences related to Chebyshev polynomials.

Cf. A000027, A000129, A049310, A053118.

Cf. A038207, A097610, A099089, A100258, A121448, A133156, A207538.

using Nemo
function A053117Row(n)
    R, x = PolynomialRing(ZZ, "x")
    p = chebyshev_u(n, x)
    [coeff(p, j) for j in 0:n] end
for n in 0:6 A053117Row(n) |> println end # Peter Luschny, Mar 13 2018

seq(seq(coeff(orthopoly[U](n,x),x,j),j=0..n),n=0..16); # Robert Israel, Feb 09 2016

Flatten[ Table[ CoefficientList[ ChebyshevU[n, x], x], {n, 0, 12}]](* Jean-François Alcover, Nov 24 2011 *)

T(n, k) = polcoeff(polchebyshev(n,2), k); \\ Michel Marcus, Feb 10 2016

a(n, m) = (2^m)*A049310(n,m).

a(n, m) := 0 if n

If n and k are of the same parity then a(n,k)=(-1)^((n-k)/2)*sum(binomial((n+k)/2,i)*binomial((n+k)/2-i,(n-k)/2),i=0..k) and a(n,k)=0 otherwise. - Milan Janjic, Apr 13 2008

A008312 Triangle of coefficients of Chebyshev polynomials U_n(x).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Version with zeros in A053117. - Philippe Deléham, Nov 27 2013

			From _Philippe Deléham_, Nov 27 2013: (Start)
Triangle begins:
    1;
    2;
   -1,    4;
   -4,    8;
    1,  -12,    16;
    6,  -32,    32;
   -1,   24,   -80,    64;
   -8,   80,  -192,   128;
    1,  -40,   240,  -448,   256;
   10, -160,   672, -1024,   512;
   -1,   60,  -560,  1792, -2304, 1024;
  -12,  280, -1792,  4608, -5120, 2048;
  ...
With zeros, triangle begins:
   1;
   0,   2;
  -1,   0,   4;
   0,  -4,   0,    8;
   1,   0, -12,    0,   16;
   0,   6,   0,  -32,    0,    32;
  -1,   0,  24,    0,  -80,     0,   64;
   0,  -8,   0,   80,    0,  -192,    0,   128;
   1,   0, -40,    0,  240,     0, -448,     0,   256;
   0,  10,   0, -160,    0,   672,    0, -1024,     0,   512;
  -1,   0,  60,    0, -560,     0, 1792,     0, -2304,     0, 1024;
   0, -12,   0,  280,    0, -1792,    0,  4608,     0, -5120,    0, 2048;
  ...
(End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.

T. D. Noe, Rows n = 0..100 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
D. Foata and G.-N. Han, Nombres de Fibonacci et polynomes orthogonaux.
Valentin Ovsienko, Towards quantized complex numbers: q-deformed Gaussian integers and the Picard group, arXiv:2103.10800 [math.QA], 2021.
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From _N. J. A. Sloane_, Feb 13 2013
Index entries for sequences related to Chebyshev polynomials.

Reflected version with zeros: A053118. Cf. A008310, A053112, A053117.

a[n_, k_] := Coefficient[ ChebyshevU[n, x], x, k]; row[n_] := Table[a[n, k], {k, Mod[n, 2], n, 2}]; Table[row[n], {n, 0, 11}] // Flatten (* Jean-François Alcover, Oct 03 2012 *)

A099089 Riordan array (1, 2+x).

Original entry on oeis.org

1, 0, 2, 0, 1, 4, 0, 0, 4, 8, 0, 0, 1, 12, 16, 0, 0, 0, 6, 32, 32, 0, 0, 0, 1, 24, 80, 64, 0, 0, 0, 0, 8, 80, 192, 128, 0, 0, 0, 0, 1, 40, 240, 448, 256, 0, 0, 0, 0, 0, 10, 160, 672, 1024, 512, 0, 0, 0, 0, 0, 1, 60, 560, 1792, 2304, 1024, 0, 0, 0, 0, 0, 0, 12, 280, 1792, 4608, 5120, 2048

Offset: 0

Paul Barry, Sep 25 2004

Row sums are A000129. Diagonal sums are A008346. The Riordan array (1, s+tx) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).
Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1/2, -1/2, 0, 0, 0, 0, ...] DELTA [2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 10 2008
As an upper right triangle (in the example), table rows give number of points, edges, faces, cubes, 4D hypercubes etc. in hypercubes of increasing dimension by column. - Henry Bottomley, Apr 14 2000. More precisely, the (i,j)-th entry is the number of j-dimensional subspaces of an i-dimensional hypercube (see the Coxeter reference). - Christof Weber, May 08 2009

			Triangle begins:
  1;
  0,  2;
  0,  1,  4;
  0,  0,  4,  8;
  0,  0,  1, 12, 16;
  0,  0,  0,  6, 32, 32;
  0,  0,  0,  1, 24, 80, 64;
The entries can also be interpreted as the antidiagonal reading of the following array:
  1,    2,    4,    8,   16,   32,   64,  128,  256,  512, 1024,... A000079
  0,    1,    4,   12,   32,   80,  192,  448, 1024, 2304, 5120,... A001787
  0,    0,    1,    6,   24,   80,  240,  672, 1792, 4608,11520,... A001788
  0,    0,    0,    1,    8,   40,  160,  560, 1792, 5376,15360,... A001789
  0,    0,    0,    0,    1,   10,   60,  280, 1120, 4032,13440,...
  0,    0,    0,    0,    0,    1,   12,   84,  448, 2016, 8064,...
  0,    0,    0,    0,    0,    0,    1,   14,  112,  672, 3360,...
  0,    0,    0,    0,    0,    0,    0,    1,   16,  144,  960,...
  0,    0,    0,    0,    0,    0,    0,    0,    1,   18,  180,...
  0,    0,    0,    0,    0,    0,    0,    0,    0,    1,   20,...
  0,    0,    0,    0,    0,    0,    0,    0,    0,    0,    1,...

H. S. M. Coxeter, Regular Polytopes, Dover Publications, New York (1973), p. 122.

Eric W. Weisstein's Mathworld, Hypercube.

Cf. A053118, A008312, A062715, A038207.

Number triangle T(n,k) = binomial(k, n-k)*2^k*(1/2)^(n-k); columns have g.f. (2*x+x^2)^k.
G.f.: 1/(1-2y*x-y*x^2). - Philippe Deléham, Nov 20 2011
Sum_ {k=0..n} T(n,k)*x^k = A000007(n), A000129(n+1), A090017(n+1), A090018(n), A190510(n+1), A190955(n+1) for x = 0,1,2,3,4,5 respectively. - Philippe Deléham, Nov 20 2011
T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = T(2,0) = 0, T(1,1) = 2, T(2,1) = 1, T(2,2) = 4, T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Oct 30 2013

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A053117 Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).

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A008312 Triangle of coefficients of Chebyshev polynomials U_n(x).

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A099089 Riordan array (1, 2+x).

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