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
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.
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.
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 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)
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
(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
(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. A053120 (increasing powers of x).
KEYWORD
AUTHOR
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
STATUS
approved