A053118 - OEIS

%I #18 Jan 27 2015 08:34:09

%S 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,

%T 128,0,-192,0,80,0,-8,0,256,0,-448,0,240,0,-40,0,1,512,0,-1024,0,672,

%U 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

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

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

%C 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).

%C Row sums (unsigned triangle) A000129(n+1) (Pell). Row sums (signed triangle) A000027(n+1) (natural numbers).

%D Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

%H T. D. Noe, <a href="/A053118/b053118.txt">Rows n=0..100 of triangle, flattened</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F 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.

%e 1;

%e 2,0;

%e 4,0,-1;

%e 8,0,-4,0;

%e 16,0,-12,0,1;

%e ... E.g. fourth row (n=3) {8,0,-4,0} corresponds to polynomial U(3,x)= 8*x^3-4*x.

%t Flatten[ Table[ Reverse[ CoefficientList[ ChebyshevU[n, x], x]], {n, 0, 12}]] (* _Jean-François Alcover_, Jan 20 2012 *)

%Y Cf. A053117, A049310, A000129.

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

%K easy,nice,sign,tabl

%O 0,2

%A _Wolfdieter Lang_