A039991 Triangle of coefficients of cos(x)^n in polynomial for cos(nx).
1, 1, 0, 2, 0, -1, 4, 0, -3, 0, 8, 0, -8, 0, 1, 16, 0, -20, 0, 5, 0, 32, 0, -48, 0, 18, 0, -1, 64, 0, -112, 0, 56, 0, -7, 0, 128, 0, -256, 0, 160, 0, -32, 0, 1, 256, 0, -576, 0, 432, 0, -120, 0, 9, 0, 512, 0, -1280, 0, 1120, 0, -400, 0, 50, 0, -1, 1024, 0, -2816, 0, 2816, 0, -1232, 0, 220, 0, -11, 0
Offset: 0
Examples
Letting c = cos x, we have: cos 0x = 1, cos 1x = 1c; cos 2x = 2c^2-1; cos 3x = 4c^3-3c, cos 4x = 8c^4-8c^2+1, etc. From _Wolfdieter Lang_, Aug 06 2014: (Start) The triangle a(n,m) begins: n\m 0 1 2 3 4 5 6 7 8 9 10 11 ... 0: 1 1: 1 0 2: 2 0 -1 3: 4 0 -3 0 4: 8 0 -8 0 1 5: 16 0 -20 0 5 0 6: 32 0 -48 0 18 0 -1 7: 64 0 -112 0 56 0 -7 0 8: 128 0 -256 0 160 0 -32 0 1 9: 256 0 -576 0 432 0 -120 0 9 0 10: 512 0 -1280 0 1120 0 -400 0 50 0 -1 11: 1024 0 -2816 0 2816 0 -1232 0 220 0 -11 0 ... -------------------------------------------------------------------------- Chebyshev T-polynomials (decreasing even or odd powers): n=3: T(3, x) = 4*x^3 - 3*x^1; n=4: T(4, x) = 8*x^4 - 8*x^2 + 1. (End)
References
- 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.
- Martin Aigner and Gunter M. Ziegler, Proofs From the Book, Springer 2004. See Chapter 18, Appendix.
- E. A. Guilleman, Synthesis of Passive Networks, Wiley, 1957, p. 593.
- Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
Links
- T. D. Noe, Table of n, a(n) for n = 0..5150
- 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].
- Daniel J. Greenhoe, Frames and Bases: Structure and Design, Version 0.20, Signal Processing ABCs series (2019) Vol. 4, see page 172.
- Daniel J. Greenhoe, A Book Concerning Transforms, Version 0.10, Signal Processing ABCs series (2019) Vol. 5, see page 94.
- Index entries for sequences related to Chebyshev polynomials.
Crossrefs
Programs
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Magma
function T(n,k) // T = A039991 if k lt 0 or k gt n then return 0; elif n lt 2 and k eq 0 then return 1; else return 2*T(n-1, k) - T(n-2, k-2); end if; return T; end function; [T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 10 2022
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Maple
seq(seq(coeff(orthopoly[T](i,x),x,i-j),j=0..i),i=0..20); # Robert Israel, Aug 07 2014
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Mathematica
row[n_] := CoefficientList[ ChebyshevT[n, x], x] // Reverse; Table[row[n], {n, 0, 11}] // Flatten(* Jean-François Alcover, Sep 14 2012 *) -
PARI
T(n,m)=(1+(-1)^m)*(binomial(n-m/2,n-m)+binomial(n-1-m/2,n-m))*2^(n-m-2)*(-1)^((m+1-(-1)^m)/2) /* Tani Akinari, Jul 18 2024 */
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SageMath
def T(n, k): # T = A039991 if (n<2 and k==0): return 1 elif (k<0 or k>n): return 0 else: return 2*T(n-1, k) - T(n-2, k-2) flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 10 2022
Formula
T(n, m) = 0 if n= 2, m >= 0; T(n, -2) = T(n, -1) = 0, T(0, 0) = T(1, 0) = 1.
G.f. for m-th column: 0 if m odd, (1-x)/(1-2*x) if m=0, else ((-1)^(m/2))*(x^m)*(1-x)/(1-2*x)^(m/2+1). For g.f. for row polynomials and row sums, see A053120.
G.f. row polynomials: (1-z)/(1 - 2*z + (x*z)^2). - Wolfdieter Lang, Aug 06 2014
Recurrence for the row polynomials Trev(n, x):= x^n*T(n, 1/x) = Sum_{m=0..n} T(n, m)*x^m; Trev(n, x) = 2*Trev(n-1, x) - x^2*Trev(n-2, x), n >= 1, Trev(-1, x) = 1/x^2 and Trev(0, x) = 1. From the T(n, x) recurrence. Compare this with A081265. - Wolfdieter Lang, Aug 07 2014
T(n,m) = (1+(-1)^m)*(binomial(n-m/2,n-m)+binomial(n-1-m/2,n-m))*2^(n-m-2)*(-1)^((m+1-(-1)^m)/2). - Tani Akinari, Jul 18 2024
Extensions
Entry improved by comments from Wolfdieter Lang, Jan 11 2000.
Edited: A053120 added in comment and crossrefs. Cfs. A028297 and A008310 specified. - Wolfdieter Lang, Aug 06 2014
Comments