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 A053120 Triangle of coefficients of Chebyshev's T(n,x) polynomials (powers of x in increasing order). 208
 1, 0, 1, -1, 0, 2, 0, -3, 0, 4, 1, 0, -8, 0, 8, 0, 5, 0, -20, 0, 16, -1, 0, 18, 0, -48, 0, 32, 0, -7, 0, 56, 0, -112, 0, 64, 1, 0, -32, 0, 160, 0, -256, 0, 128, 0, 9, 0, -120, 0, 432, 0, -576, 0, 256, -1, 0, 50, 0, -400, 0, 1120, 0, -1280, 0, 512, 0, -11, 0, 220, 0, -1232, 0, 2816, 0, -2816, 0, 1024 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Row sums (signed triangle): A000012 (powers of 1). Row sums (unsigned triangle): A001333(n). From Wolfdieter Lang, Oct 21 2013: (Start) The row polynomials T(n,x) equal (S(n,2*x) - S(n-2,2*x))/2, n >= 0, with the row polynomials S from A049310, with S(-1,x) = 0, and S(-2,x) = -1. The zeros of T(n,x) are x(n,k) = cos((2*k+1)*Pi/(2*n)), k = 0, 1, ..., n-1, n >= 1. (End) From Wolfdieter Lang, Jan 03 2020 and Paul Weisenhorn: (Start) The (sub)diagonal sequences {D_{2*k}(m)}_{m >= 0}, for k >= 0, have o.g.f. GD_{2*k}(x) = (-1)^k*(1-x)/(1-2*x)^(k+1), for k >= 0, and GD_{2*k+1}(x) = 0, for k >= 0. This follows from their o.g.f. GGD(z, x) := Sum_{k>=0} GD_k(x)*z^n which is obtained from the o.g.f of the T-triangle GT(z, x) = (1-x*z)/(1 - 2*x + z^2) (see the formula section) by GGD(z, x) = GT(z, x/z). The explicit form is then D_{2*k}(m) = (-1)^k, for m = 0, and (-1)^k*(2*k+m)*2^(m-1)*risefac(k+1, m-1)/m!, for m >= 1, with the rising factorial risefac(x, n). (End) REFERENCES M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964. Tenth printing, Wiley, 2002 (also electronically available), p. 795. F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg 1994 pp. 77, 105. Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990. TableCurve 2D, Automated curve fitting and equation discovery, Version 5.01 for Windows, User's Manual, Chebyshev Series Polynomials and Rationals, pages 12-21 - 12-24, SYSTAT Software, Inc., Richmond, WA, 2002. LINKS T. D. Noe, Rows 0 to 100 of triangle, flattened M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [scanned copy], p.795. Paul Barry and A. Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, J. Int. Seq. 13 (2010) # 10.9.4, section 5. Tom Copeland, Addendum to Elliptic Lie Triad P. Damianou, On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.- From Tom Copeland, Oct 11 2014 Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011. Wolfdieter Lang, Rows n = 0..20. Wikipedia, Chebyshev polynomials FORMULA T(n, m) = A039991(n, n-m). G.f. for row polynomials T(n,x) (signed triangle): (1-x*z)/(1-2*x*z+z^2). If unsigned: (1-x*z)/(1-2*x*z-z^2). T(n, m) := 0 if n < m or n+m odd; T(n, m) = (-1)^(n/2) if m=0 (n even); otherwise T(n, m) = ((-1)^((n+m)/2 + m))*(2^(m-1))*n*binomial((n+m)/2-1, m-1)/m. Recursion for n >= 2: T(n, m) = T*a(n-1, m-1) - T(n-2, m), T(n, m)=0 if n < m, T(n, -1) := 0, T(0, 0) = T(1, 1) = 1. G.f. for m-th column (signed triangle): 1/(1+x^2) if m=0, otherwise (2^(m-1))*(x^m)*(1-x^2)/(1+x^2)^(m+1). From G. C. Greubel, Aug 10 2022: (Start) Sum_{k=0..floor(n/2)} T(n-k, k) = A000007(n). T(2*n, n) = i^n * A036909(n/2) * (1+(-1)^n)/2 + [n=0]/3. (End) T(n, k) = [x^k] T(n, x) for n >= 1, where T(n, x) = Sum_{k=1..n}(-1)^(n - k)*(n/ (2*k))*binomial(k, n - k)*(2*x)^(2*k - n). - Peter Luschny, Sep 20 2022 EXAMPLE The triangle a(n,m) begins: n\m 0 1 2 3 4 5 6 7 8 9 10... 0: 1 1: 0 1 2: -1 0 2 3: 0 -3 0 4 4: 1 0 -8 0 8 5: 0 5 0 -20 0 16 6: -1 0 18 0 -48 0 32 7: 0 -7 0 56 0 -112 0 64 8: 1 0 -32 0 160 0 -256 0 128 9: 0 9 0 -120 0 432 0 -576 0 256 10: -1 0 50 0 -400 0 1120 0 -1280 0 512 ... Reformatted and extended - Wolfdieter Lang, Oct 21 2013 E.g., the fourth row (n=3) corresponds to the polynomial T(3,x) = -3*x + 4*x^3. MAPLE with(orthopoly) ; A053120 := proc(n, k) T(n, x) ; coeftayl(%, x=0, k) ; end proc: # R. J. Mathar, Jun 30 2013 T := (n, x) -> `if`(n = 0, 1, add((-1)^(n - k) * (n/(2*k))*binomial(k, n - k) *(2*x)^(2*k - n), k = 1 ..n)): seq(seq(coeff(T(n, x), x, k), k = 0..n), n = 0..11); # Peter Luschny, Sep 20 2022 MATHEMATICA t[n_, k_] := Coefficient[ ChebyshevT[n, x], x, k]; Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (* Jean-François Alcover, Jan 16 2012 *) PROG (Magma) &cat[ Coefficients(ChebyshevT(n)): n in [0..11] ]; // Klaus Brockhaus, Mar 08 2008 (PARI) for(n=0, 5, P=polchebyshev(n); for(k=0, n, print1(polcoeff(P, k)", "))) \\ Charles R Greathouse IV, Jan 16 2012 (Julia) using Nemo function A053120Row(n) R, x = PolynomialRing(ZZ, "x") p = chebyshev_t(n, x) [coeff(p, j) for j in 0:n] end for n in 0:6 A053120Row(n) |> println end # Peter Luschny, Mar 13 2018 (SageMath) def f(n, k): # f = A039991 if (n<2 and k==0): return 1 elif (k<0 or k>n): return 0 else: return 2*f(n-1, k) - f(n-2, k-2) def A053120(n, k): return f(n, n-k) flatten([[A053120(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 10 2022 CROSSREFS Cf. A000007, A000012, A001333, A036909, A039991. The first nonzero (sub)diagomal sequences are A011782, -A001792, A001793(n+1), -A001794, A006974, -A006975, A006976, -A209404. Sequence in context: A223707 A046767 A115720 * A336836 A284976 A008743 Adjacent sequences: A053117 A053118 A053119 * A053121 A053122 A053123 KEYWORD sign,tabl,nice,easy AUTHOR STATUS approved

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Last modified February 5 18:47 EST 2023. Contains 360087 sequences. (Running on oeis4.)