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A127672
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Monic integer version of Chebyshev T-polynomials (increasing powers).
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8
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2, 0, 1, -2, 0, 1, 0, -3, 0, 1, 2, 0, -4, 0, 1, 0, 5, 0, -5, 0, 1, -2, 0, 9, 0, -6, 0, 1, 0, -7, 0, 14, 0, -7, 0, 1, 2, 0, -16, 0, 20, 0, -8, 0, 1, 0, 9, 0, -30, 0, 27, 0, -9, 0, 1, -2, 0, 25, 0, -50, 0, 35, 0, -10, 0, 1, 0, -11, 0, 55, 0, -77, 0, 44, 0, -11, 0, 1, 2, 0, -36, 0, 105, 0, -112, 0, 54, 0, -12, 0, 1, 0, 13, 0, -91
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| The row polynomials R(n,x):=sum(a(n,m)*x^m),m=0..n) have been called Chebyshev C_n(x) polynomials in the Abramowitz-Stegun handbook, p. 778, 22.5.11
(see A049310 for the reference, and note that on p. 774 the S and C polynomials have been mixed up in older printings). [From W. Lang, Jun 03 2011]
This is a signed version of triangle A114525.
The unsigned column sequences (without zeros) are, for m=1..11: A005408, A000290, A000330, A002415, A005585, A040977, A050486, A053347, A054333, A054334, A057788.
The row polynomials R(n,x):= sum(a(n,m)*x*m,m=0..n), give for n=2,3,...,floor(N/2) the positive zeros of the Chebyshev S(N-1,x)-polynomial (see A049310) in terms of its largest zero rho(N):= 2*cos(Pi/N) by putting x=rho(N). The order of the positive zeros is falling: n=1 corresponds to the largest zero rho(N) and n=floor(N/2) to the smallest positive zero. Example N=5: rho(5)=phi (golden section), R(2,phi)= phi^2-2 = phi-1, the second largest (and smallest) positive zero of S(4,x). W. Lang, Dec 01 2010.
The row polynomial R(n,x), for n>=1, factorizes into minimal polynomials of 2*cos(pi/k), called C(k,x), with coefficients given in A187360, as follows.
R(n,x) = product(C(2*n/d,x),d|oddpart(n))
= product(C(2^{k+1}*d,x),d|oddpart(n)),
with oddpart(n)=A000265(n), and 2^k is the largest power of 2 dividing n, where k=0,1,2,...
(Proof: R and C are monic, the degree on both sides coincides, and the zeros of R(n,x) appear all on the r.h.s.). [From W. Lang, Jul 31 2011]
The zeros of the row polynomials R(n,x) are 2*cos(Pi*(2*k+1)/(2*n)), k=0,1, ..., n-1; n>=1 (from those of the Chebyshev T-polynomials). [From W. Lang, Sep 17 2011]
The discriminants of the row polynomials R(n,x) are found under A193678. [From W. Lang, Aug 27 2011]
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LINKS
| W. Lang, Row polynomials.
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FORMULA
| a(n,0)=0 if n is odd, a(n,0)=2*(-1)^(n/2) if n is even, else a(n,m)=t(n,m)/2^(m-1) with t(n,m):=A053120(n,m) (coefficients of Chebyshev T-polynomials).
G.f. for m-th column (signed triangle): 2/(1+x^2) if m=0 else (x^m)*(1-x^2)/(1+x^2)^(m+1).
Riordan type matrix ((1-x^2)/(1+x^2),x/(1+x^2)) if one puts a(0,0)=1 (instead of 2).
O.g.f. for row polynomials: R(x,z):=sum(n>=0, R(n,x)*z^n )=(2-x*z)*S(x,z), with the o.g.f. S(x,z)= 1/(1-x*z+z^2) for the S-polynomials (see A049310).
Note that R(n,x) = R(2*n,sqrt(2+x)). n>=0 (from the o.g.f.s of both sides).
[From W. Lang Jun 03 2011]
a(n,m) := 0 if n<m or n+m odd; a(n,0)= 2*(-1)^(n/2) (n even); else a(n,m)=((-1)^((n+m)/ 2+m))*n*binomial((n+m)/2-1,m-1)/m.
Recursion for n >= 2 and m>=2: a(n,m) = a(n-1,m-1)-a(n-2,m), a(n,m)=0 if n<m, a(2*k,1)=0, a(2*k+1,1)=(2*k+1)*(-1)^k. In addition, for column m=0: a(2*k,0)= 2*(-1)^k, a(2*k+1,0)=0, k>=0.
Chebyshev T(n,x) = sum(m=0..n, a(n,m)*(2^(m-1))*x^m ). [From W. Lang, Jun 03 2011]
R(n,x) = 2*T(n,x/2) = S(n,x) - S(n-2,x), n>=0, with Chebyshev's T- and S-polynomials, showing that they are integer and monic polynomials. - From Wolfdieter Lang, Nov 08 2011.
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EXAMPLE
| Row n=4: [2,0,-4,0,1] stands for the polynomial 2*y^0 - 4*y^2 + 1*y^4. With y^m replaced by 2^(m-1)*x^m this becomes T(4,x)= 1-8*x^2+8*x^4.
Triangle begins:
n\m 0 1 2 3 4 5 6 7 8 9 10 ...
0: 2
1: 0 1
2: -2 0 1
3: 0 -3 0 1
4: 2 0 -4 0 1
5: 0 5 0 -5 0 1
6: -2 0 9 0 -6 0 1
7: 0 -7 0 14 0 -7 0 1
8: 2 0 -16 0 20 0 -8 0 1
9: 0 9 0 -30 0 27 0 -9 0 1
10: -2 0 25 0 -50 0 35 0 -10 0 1
...
Factorization into minimal C-polynomials:
R(12,x) = R((2^2)*3,x) = C(24,x)*C(8,x) = C((2^3)*1,x)*C((2^3)*3,x). [From W. Lang, Jul 31 2011]
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CROSSREFS
| Row sums (signed): A057079(n-1). Row sums (unsigned): A000032(n) (Lucas numbers).
Bisection: A127677(even n triangle, without zero entries), ((-1)^(n-m))*A111125(n, m) (odd n triangle, without zero entries).
Cf. A108045.
Sequence in context: A115723 A178524 A114525 * A166124 A134979 A112248
Adjacent sequences: A127669 A127670 A127671 * A127673 A127674 A127675
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KEYWORD
| sign,tabl,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Mar 07 2007
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EXTENSIONS
| Name changed and table rewritten by Wolfdieter Lang, Nov 08 2011.
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