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 A053119 Triangle of coefficients of Chebyshev's S(n,x) polynomials (exponents in decreasing order). 9
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 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 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 T. Copeland, Addendum to Elliptic Lie Triad 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, F=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 EXAMPLE 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:  [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 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 *) PROG (PARI) tabl(nn) = for (n=0, nn, print(Vec(polchebyshev(n, 2, x/2)))); \\ Michel Marcus, Jan 14 2016 CROSSREFS Row sums give A000045. Reflection of A049310. Cf. A162515. - Wolfdieter Lang, Dec 16 2013 Sequence in context: A225345 A083280 A060689 * A162515 A175267 A108045 Adjacent sequences:  A053116 A053117 A053118 * A053120 A053121 A053122 KEYWORD easy,nice,sign,tabl AUTHOR STATUS approved

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Last modified March 30 16:16 EDT 2020. Contains 333127 sequences. (Running on oeis4.)