A053118 Triangle of coefficients of Chebyshev's U(n,x) polynomials (exponents in decreasing order).

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

Offset

0,2

Comments

a(n,m)= A053117(n,n-m) = 2^(n-m)*A049310(n,n-m).

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).

References

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

Index entries for sequences related to Chebyshev polynomials.

Formula

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

Example

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

Mathematica

Flatten[ Table[ Reverse[ CoefficientList[ ChebyshevU[n, x], x]], {n, 0, 12}]] (* Jean-François Alcover, Jan 20 2012 *)

Crossrefs

Cf. A053117, A049310, A000129.

Triangle reflected without zeros: A008312 (the main entry).

Sequence in context: A051517 A289359 A356771 * A175682 A326722 A349127

Adjacent sequences: A053115 A053116 A053117 * A053119 A053120 A053121

Keyword

easy,nice,sign,tabl

Author

Wolfdieter Lang

Status

approved