0,9

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).

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.

T. D. Noe, Rows n=0..100 of triangle, flattened

Index entries for sequences related to Chebyshev polynomials.

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].

{1}; {1,0}; {1,0,-1}; {1,0,-2,0}; {1,0,-3,0,1};... 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]

Row sums give A000045.

Reflection of A049310.

Sequence in context: A007814 A083280 A060689 this_sequence A162515 A108045 A143728

Adjacent sequences: A053116 A053117 A053118 this_sequence A053120 A053121 A053122

easy,nice,sign,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)