login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A127672 Monic integer version of Chebyshev T-polynomials (increasing powers). 47

%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). - _Wolfdieter 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). - _Wolfdieter 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.). - _Wolfdieter 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). - _Wolfdieter Lang_, Sep 17 2011

%C The discriminants of the row polynomials R(n,x) are found under A193678. - _Wolfdieter Lang_, Aug 27 2011

%C 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

%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

%C From _Tom Copeland_, Nov 08 2015: (Start)

%C 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.

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

%C 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.)

%C 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).

%C (End)

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

%H Robert Israel, <a href="/A127672/b127672.txt">Table of n, a(n) for n = 0..10010</a> (rows 0 to 140, flattened)

%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 T. Copeland, <a href="http://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">Addendum to Elliptic Lie Triad</a>

%H P. Damianou , <a href="http://arxiv.org/abs/1110.6620">On the characteristic polynomials of Cartan matrices and Chebyshev polynomials</a>, arXiv preprint arXiv:1110.6620 [math.RT], 2014.

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

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev 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). - _Wolfdieter 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 ). - _Wolfdieter 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

%F From _Tom Copeland_, Nov 08 2015: (Start)

%F 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.

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

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

%F (End)

%F 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

%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

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

%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.

%Y Cf. A263916.

%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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 5 19:19 EST 2016. Contains 278770 sequences.