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A127672 Monic integer version of Chebyshev T-polynomials (increasing powers). 34

%I

%S 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,

%T 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,

%U -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

%N Monic integer version of Chebyshev T-polynomials (increasing powers).

%C 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

%C (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]

%C This is a signed version of triangle A114525.

%C The unsigned column sequences (without zeros) are, for m=1..11: A005408, A000290, A000330, A002415, A005585, A040977, A050486, A053347, A054333, A054334, A057788.

%C 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.

%C 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.

%C R(n,x) = product(C(2*n/d,x),d|oddpart(n))

%C = product(C(2^{k+1}*d,x),d|oddpart(n)),

%C with oddpart(n)=A000265(n), and 2^k is the largest power of 2 dividing n, where k=0,1,2,...

%C (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]

%C 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]

%C The discriminants of the row polynomials R(n,x) are found under A193678. [From W. Lang, Aug 27 2011]

%C The determinant of the NxN 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

%C 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

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

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Wolfdieter Lang, <a href="/A127672/a127672.pdf">Row polynomials.</a>

%F 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).

%F 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).

%F Riordan type matrix ((1-x^2)/(1+x^2),x/(1+x^2)) if one puts a(0,0)=1 (instead of 2).

%F 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).

%F Note that R(n,x) = R(2*n,sqrt(2+x)). n>=0 (from the o.g.f.s of both sides).

%F [From W. Lang Jun 03 2011]

%F 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.

%F 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.

%F Chebyshev T(n,x) = sum(m=0..n, a(n,m)*(2^(m-1))*x^m ). [From W. Lang, Jun 03 2011]

%F 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.

%e 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.

%e Triangle begins:

%e n\m 0 1 2 3 4 5 6 7 8 9 10 ...

%e 0: 2

%e 1: 0 1

%e 2: -2 0 1

%e 3: 0 -3 0 1

%e 4: 2 0 -4 0 1

%e 5: 0 5 0 -5 0 1

%e 6: -2 0 9 0 -6 0 1

%e 7: 0 -7 0 14 0 -7 0 1

%e 8: 2 0 -16 0 20 0 -8 0 1

%e 9: 0 9 0 -30 0 27 0 -9 0 1

%e 10: -2 0 25 0 -50 0 35 0 -10 0 1 ...

%e Factorization into minimal C-polynomials:

%e 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]

%Y Row sums (signed): A057079(n-1). Row sums (unsigned): A000032(n) (Lucas numbers).

%Y Bisection: A127677 (even n triangle, without zero entries), ((-1)^(n-m))*A111125(n, m) (odd n triangle, without zero entries).

%Y Cf. A108045.

%K sign,tabl,easy

%O 0,1

%A _Wolfdieter Lang_ Mar 07 2007

%E Name changed and table rewritten by _Wolfdieter Lang_, Nov 08 2011.

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Last modified April 25 04:22 EDT 2014. Contains 240994 sequences.