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A053123
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Triangle of coefficients of shifted Chebyshev's S(n,x-2)= U(n,x/2-1) polynomials (exponents of x in decreasing order).
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17
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1, 1, -2, 1, -4, 3, 1, -6, 10, -4, 1, -8, 21, -20, 5, 1, -10, 36, -56, 35, -6, 1, -12, 55, -120, 126, -56, 7, 1, -14, 78, -220, 330, -252, 84, -8, 1, -16, 105, -364, 715, -792, 462, -120, 9, 1, -18, 136, -560, 1365, -2002, 1716, -792, 165, -10, 1, -20, 171, -816, 2380, -4368, 5005, -3432, 1287, -220, 11, 1
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OFFSET
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0,3
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COMMENTS
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G.f. for row polynomials and row sums same as in A053122.
This is also the coefficient triangle for Chebyshev's U(2*n+1,x) polynomials expanded in decreasing odd powers of (2*x): U(2*n+1,x) = Sum_{m=0..n} (T(n,m)*(2*x)^(2*(n-m)+1). See the W. Lang link given in A053125.
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
Stephen Barnett, "Matrices: Methods and Applications", Oxford University Press, 1990, p. 132, 343.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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T(n, m) = 0 if n<m else (-1)^m*binomial(2*n+1-m, m);
T(n, m) = -2*T(n-1, m-1) + T(n-1, m) - T(n-2, m-2), T(n, -2) = 0, T(-2, m) = 0, T(n, -1) = 0 = T(-1, m), T(0, 0)=1, T(n, m)= 0 if n<m.
G.f. for m-th column (signed triangle): ((-1)^m)*x^m*Po(m+1, x)/(1-x)^(m+1), with Po(k, x) := sum('binomial(k, 2*j+1)*x^j', 'j'=0..floor(k/2)).
The n-th degree polynomial is the characteristic equation for an n X n tridiagonal matrix with (diagonal = all 2's, sub and superdiagonals all -1's and the rest 0's), exemplified by the 4X4 matrix M = [2 -1 0 0 / -1 2 -1 0 / 0 -1 2 -1 / 0 0 -1 2]. - Gary W. Adamson, Jan 05 2005
sum(m=0..n, a(n,m)*(c(n))^(2*n-2*m) ) = 1/c(n), where c(n) = 2*cos(Pi/(2*n+3)). - L. Edson Jeffery, Sep 13 2013
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EXAMPLE
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Triangle begins:
1;
1, -2;
1, -4, 3;
1, -6, 10, -4;
1, -8, 21, -20, 5;
1, -10, 36, -56, 35, -6;
1, -12, 55, -120, 126, -56, 7; ...
E.g. fourth row (n=3) {1,-6,10,-4} corresponds to polynomial S(3,x-2)= x^3-6*x^2+10*x-4.
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MAPLE
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(-1)^m*binomial(2*n+1-m, m) ;
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MATHEMATICA
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T[n_, m_]:= (-1)^m*Binomial[2*n+1-m, m]; Table[T[n, m], {n, 0, 11}, {m, 0, n}]//Flatten (* Jean-François Alcover, Mar 05 2014, after R. J. Mathar *)
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PROG
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(PARI) for(n=0, 10, for(k=0, n, print1((-1)^k*binomial(2*n-k+1, k), ", "))) \\ G. C. Greubel, Jul 23 2019
(Magma) [(-1)^k*Binomial(2*n-k+1, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jul 23 2019
(Sage) [[(-1)^k*binomial(2*n-k+1, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jul 23 2019
(GAP) Flat(List([0..10], n-> List([0..n], k-> (-1)^k*Binomial(2*n-k+1, k) ))); # G. C. Greubel, Jul 23 2019
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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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