A053119 Triangle of coefficients of Chebyshev's S(n,x) polynomials (exponents in decreasing order).

1, 1, 0, 1, 0, -1, 1, 0, -2, 0, 1, 0, -3, 0, 1, 1, 0, -4, 0, 3, 0, 1, 0, -5, 0, 6, 0, -1, 1, 0, -6, 0, 10, 0, -4, 0, 1, 0, -7, 0, 15, 0, -10, 0, 1, 1, 0, -8, 0, 21, 0, -20, 0, 5, 0, 1, 0, -9, 0, 28, 0, -35, 0, 15, 0, -1, 1, 0, -10, 0, 36, 0, -56, 0, 35, 0, -6, 0, 1, 0, -11, 0, 45, 0, -84, 0, 70, 0, -21, 0, 1

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nice
• sign
• tabl

Comments

These polynomials also give the determinant of the tridiagonal matrix having x on the diagonal and -1 next to these x. - M. F. Hasler, Oct 15 2019

The polynomial S(n,x) is the character of the irreducible (n+1) dimensional representation of the Lie algebra sl_2 when x is the character of irreducible 2-dimesional representation. - Leonid Bedratyuk, Oct 28 2023

Examples

			The triangle begins:
n\m 0  1   2  3   4  5   6  7   8  9  10 ...
0:  1
1:  1  0
2:  1  0  -1
3:  1  0  -2  0
4:  1  0  -3  0   1
5:  1  0  -4  0   3  0
6:  1  0  -5  0   6  0  -1
7:  1  0  -6  0  10  0  -4  0
8:  1  0  -7  0  15  0 -10  0   1
9:  1  0  -8  0  21  0 -20  0   5  0
10: 1  0  -9  0  28  0 -35  0  15  0  -1
... Reformatted. - _Wolfdieter Lang_, Dec 17 2013
E.g., fourth row (n=3) corresponds to polynomial S(3,x)= x^3-2*x.
Triangle of absolute values of coefficients (coefficients of Fibonacci polynomials) with exponents in increasing order begins:
[1]
[0, 1]
[1, 0, 1]
[0, 2, 0, 1]
[1, 0, 3, 0, 1]
[0, 3, 0, 4, 0, 1]
[1, 0, 6, 0, 5, 0, 1]
[0, 4, 0, 10, 0, 6, 0, 1]
[1, 0, 10, 0, 15, 0, 7, 0, 1]
[0, 5, 0, 20, 0, 21, 0, 8, 0, 1]
See A162515 for the Fibonacci polynomials with reversed row entries, starting there with row 1. - _Wolfdieter Lang_, Dec 16 2013
		

References

• D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, 1970; p. 232, Sect. 3.3.38.
• Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

Links

• T. D. Noe, Rows n=0..100 of triangle, flattened
• Tom Copeland, Addendum to Elliptic Lie Triad
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Row sums give A000045. Reflection of A049310.

Cf. A162515. - Wolfdieter Lang, Dec 16 2013

Programs

Maple

A053119 := (n, k) -> if k::even then (-1)^binomial(k, 2)*binomial(n - k/2, k/2)
else 0 fi: seq(seq(A053119(n, k), k = 0..n), n = 0..11); # Peter Luschny, Jul 20 2024

Mathematica

ChebyshevS[n_, x_] := ChebyshevU[n, x/2]; Flatten[ Table[ Reverse[ CoefficientList[ ChebyshevS[n, x], x]], {n, 0, 12}]] (* Jean-François Alcover, Nov 25 2011 *)

PARI

tabl(nn) = for (n=0, nn, print(Vec(polchebyshev(n, 2, x/2)))); \\ Michel Marcus, Jan 14 2016

Formula

a(n,m) = A049310(n,n-m).

G.f. for row polynomials S(n,x) (signed triangle): 1/(1-x*z+z^2).

Unsigned triangle |a(n,m)| has Fibonacci polynomials F(n+1,x) as row polynomials with G.f. 1/(1-x*z-z^2).

a(n, m) := 0 if n < m or m odd, else ((-1)^(3*m/2))*binomial(n-m/2, n-m); a(n, m) = a(n-1, m) - a(n-2, m-2), a(n, -2) := 0 =: a(n, -1), a(0, 0) = 1, a(n, m) = 0 if n < m or m odd.

G.f. for m-th column (signed triangle): (-1)^(3*m/2)*x^m/(1-x)^(m/2+1) if m >= 0 is even else 0.

Recurrence for the (unsigned) Fibonacci polynomials: F[1]=1, F[2]=x; for n>2, F[n] = x*F[n-1]+F[n-2].

a = 2*A192011 - 3*A192174. - Thomas Baruchel, Jun 02 2018

Recurrence for the polynomials S(n) = x S(n-1) - S(n-2); S(0) = 1, S(1) = x. - M. F. Hasler, Oct 15 2019