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A053119
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Triangle of coefficients of Chebyshev's S(n,x) polynomials (exponents in decreasing order).
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12
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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
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OFFSET
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0,9
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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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].
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
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EXAMPLE
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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
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]
SeeA162515for the Fibonacci polynomials with reversed row entries, starting there with row 1. -Wolfdieter Lang,Dec 16 2013
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MAPLE
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A053119:= (n, k) -> if k::even then (-1)^binomial(k, 2)*binomial(n - k/2, k/2)
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MATHEMATICA
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ChebyshevS[n_, x_]:= ChebyshevU[n, x/2]; Flatten[ Table[ Reverse[ CoefficientList[ ChebyshevS[n, x], x]], {n, 0, 12}]] (*Jean-François Alcover,Nov 25 2011 *)
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PROG
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(PARI) tabl(nn) = for (n=0, nn, print(Vec(polchebyshev(n, 2, x/2)))); \\Michel Marcus,Jan 14 2016
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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