|
|
A039991
|
|
Triangle of coefficients of cos(x)^n in polynomial for cos(nx).
|
|
18
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
Also triangle of coefficients of Chebyshev polynomials of first kind (T(n,x)) in decreasing order of powers of x. A053120 gives the coefficients in increasing order.
The polynomials R(n,x) := Sum_{m=0..n} a(n,m)*sqrt(x)^m, have g.f. (1-z)/(1 - 2*z + x*z^2) = ((1-z)/(1-2*z))/(1 - x*(-z^2/(1-2*z))) (from the row reversion of the g.f. of A053120 and x^2 -> x). Therefore this triangle becomes the Riordan triangle ((1-z)/(1-2*z), -z^2/(1-2*z)) if the vanishing columns are deleted (see A028297) and zeros are appended in each row numbered n>=1 in order to obtain a triangle. This is then A201701 with negative odd numbered columns. - Wolfdieter Lang, Aug 06 2014
|
|
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
|
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].
|
|
FORMULA
|
T(n, m) = 0 if n<m or m odd, (-1)^(m/2) if m=n is even, ((-1)^(3*m/2))*(2^(n-m-1))*n*binomial(n-1-m/2, n-1-m)/(n-m) else T(n, m) = 2*T(n-1, m) - T(n-2, m-2), 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
|
|
EXAMPLE
|
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.
The triangle a(n,m) begins:
n\m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
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
12: 2048 0 -6144 0 6912 0 -3584 0 840 0 -72 0 1
13: 4096 0 -13312 0 16640 0 -9984 0 2912 0 -364 0 13 0
14: 8192 0 -28672 0 39424 0 -26880 0 9408 0 -1568 0 98 0 -1
15: 16384 0 -61440 0 92160 0 -70400 0 28800 0 -6048 0 560 0 -15 0
...
--------------------------------------------------------------------------
Chebyshev T-polynomials (decreasing even or odd powers):
n=3: T(3, n) = 4*x^3 - 3*x^1; n=4: T(4, x) = 8*x^4 - 8*x^2 + 1. (End)
|
|
MAPLE
|
seq(seq(coeff(orthopoly[T](i, x), x, i-j), j=0..i), i=0..20); # Robert Israel, Aug 07 2014
|
|
MATHEMATICA
|
row[n_] := CoefficientList[ ChebyshevT[n, x], x] // Reverse; Table[row[n], {n, 0, 11}] // Flatten(* Jean-François Alcover, Sep 14 2012 *)
|
|
PROG
|
(Magma)
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
(SageMath)
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
(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 */
|
|
CROSSREFS
|
Cf. A028297 (without vanishing columns). A008310 (zero columns deleted then rows reversed).
Cf. A053120 (increasing powers of x).
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|