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A127672 Monic integer version of Chebyshev T-polynomials (increasing powers). 47
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; text; internal format)
OFFSET

0,1

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). - Wolfdieter 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). - Wolfdieter 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.). - Wolfdieter 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). - Wolfdieter Lang, Sep 17 2011

The discriminants of the row polynomials R(n,x) are found under A193678. - Wolfdieter Lang, Aug 27 2011

The determinant of the N x N matrix M(N) with entries M(N;n,m) = R(m-1,x[n]), 1 <= n,m <= N, N>=1, and any x[n], is identical with twice the Vandermondian Det(V(N)) with matrix entries V(N;n,m) = x[n]^(m-1). This is an instance of the general theorem given in the Vein-Dale reference on p. 59. Note that R(0,x) = 2 (not 1). See also the comments from Aug 26 2013 under A049310 and from Aug 27 2013 under A000178. - Wolfdieter Lang, Aug 27 2013

This triangle a(n,m) is also used to express in the regular (2*(n+1))-gon, inscribed in a circle of radius R, the length ratio side/R, called s(2*(n+1)), as a polynomial in rho(2*(n+1)), the length ratio (smallest diagonal)/side. See the bisections ((-1)^(k-s))*A111125(k,s) and A127677 for comments and examples. - Wolfdieter Lang, Oct 05 2013

From Tom Copeland, Nov 08 2015: (Start)

These are the characteristic polynomials a_n(x) = 2 T_n(x/2) for the adjacency matrix of the Coxeter simple Lie algebra  B_n, related to the Cheybshev polynomials of the first kind, T_n(x) = cos(n*q) with x = cos(q) (see p. 20 of Damianou).  Given the polynomial (x - t)(x - 1/t) = 1 -(t + 1/t) x + x^2 = e2 - e1 x + x^2, the symmetric power sums p_n(t,1/t) = t^n + t^(-n) of the zeros of this polynomial may be expressed in terms of the elementary symmetric polynomials e1 = t + 1/t = y and e2 = t*1/t = 1 as p_n(t,1/t) = a_n(y) = F(n,-y,1,0,0,..), where F(n,b1,b2,..,bn) are the Faber polynomials of A263916.

The partial sum of the first n+1 rows given t and y = t + 1/t is PS(n,t) =

sum(k=0 to n, a_n(y)) = [t^(n/2) + t^(-n/2)][t^((n+1)/2) - t^(-(n+1)/2)] / [t^(1/2) - t^(-1/2)]. (For n prime, this is related simply to the cyclotomic polynomials.)

Then a_n(y) = PS(n,t) - PS(n-1,t), and for t = e^(iq), y = 2 cos(q), and, therefore,  a_n(2 cos(q)) =  PS(n,e^(iq)) - PS(n-1,e^(iq)) = 2 cos(nq) = 2 T_n(cos(q)) with PS(n,e^(iq)) = 2 cos(nq/2) sin((n+1)q/2) / sin(q/2).

(End)

REFERENCES

R. Vein and P. Dale, Determinants and Their Applications in Mathematical Physics, Springer, 1999.

LINKS

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)

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].

T. 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.

Wolfdieter Lang, Row polynomials.

Index entries for sequences related to Chebyshev polynomials.

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). - Wolfdieter 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 ). - Wolfdieter 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. - Wolfdieter Lang, Nov 08 2011

From Tom Copeland, Nov 08 2015: (Start)

a(n,x)  = sqrt[2 + a(2n,x)], or 2 + a(2n,x) = (a(n,x))^2, is a reflection of the relation of the Chebyshev polynomials of the first kind to the cosine and the half-angle formula, (cos(q/2))^2 = (1 + cos(q))/2.

Examples: For n = 2, -2 + x^2 = sqrt(2 + 2 - 4 x^2 + x^4).

For n = 3, -3 x + x^3 =  sqrt(2 - 2 + 9 x^2 - 6 x^4 + x^6).

(End)

L(x,h1,h2) = -log(1 - h1*x + h2*x^2) = Sum_{n>0} F(n,-h1,h2,0,..,0) x^n/n =  h1 x + (-2h2 + h1^2) x^2/2 + (-3h1 h2 + h1^3) x^3/3 + .... is a log series generator of the bivariate row polynomials where T(0,0) = 0 and F(n,b1,b2,..,bn) are the Faber polynomials of A263916. Exp(L(x,h1,h2)) = 1 / (1 - h1*x + h2*x^2 ) is the o.g.f. of A049310. - Tom Copeland, Feb 15 2016

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). - Wolfdieter Lang, Jul 31 2011

MAPLE

seq(seq(coeff(2*orthopoly[T](n, x/2), x, j), j=0..n), n=0..20); # Robert Israel, Aug 04 2015

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.

Cf. A263916.

Sequence in context: A178524 A212357 A114525 * A227698 A166124 A134979

Adjacent sequences:  A127669 A127670 A127671 * A127673 A127674 A127675

KEYWORD

sign,tabl,easy

AUTHOR

Wolfdieter Lang, Mar 07 2007

EXTENSIONS

Name changed and table rewritten by Wolfdieter Lang, Nov 08 2011

STATUS

approved

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Last modified March 24 06:12 EDT 2017. Contains 283984 sequences.