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 A127672 Monic integer version of Chebyshev T-polynomials (increasing powers). 74

%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 [Theorem 1B, eq. (43) in the W. Lang link. - _Wolfdieter Lang_, Apr 13 2018]

%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)

%C R(45, x) is the famous polynomial used by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593 to pose four problems, solved by Viète. See e.g., the Havil reference, pp. 69-74. - _Wolfdieter Lang_, Apr 28 2018

%C From _Wolfdieter Lang_, May 05 2018: (Start)

%C Some identities for the row polynomials R(n, x) following from the known ones for Chebyshev T-polynomials (A053120) are:

%C (1) R(-n, x) = R(n, x).

%C (2) R(n*m, x) = R(n, R(m, x)) = R(m, R(n, x)).

%C (3) R(2*k+1, x) = (-1)^k*x*S(2*k, sqrt(4-x^2)), k >= 0, with the S row polynomials of A049310.

%C (4) R(2*k, x) = R(k, x^2-2), k >= 0.

%C (End)

%D Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.

%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.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972.

%H P. Damianou , <a href="https://arxiv.org/abs/1110.6620">On the characteristic polynomials of Cartan matrices and Chebyshev polynomials</a>, arXiv preprint arXiv:1110.6620 [math.RT], 2011-2014.

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

%H Wolfdieter Lang, <a href="http://arxiv.org/abs/1210.1018">The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon</a>, arXiv:1210.1018 [math.GR], 2012-2017.

%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

%t a[n_, k_] := SeriesCoefficient[(2 - t*x)/(1 - t*x + x^2), {x, 0, n}, {t, 0, k}]; Flatten[Table[a[n, k], {n, 0, 12}, {k, 0, n}]] (* _L. Edson Jeffery_, Nov 02 2017 *)

%Y Row sums (signed): A057079(n-1). Row sums (unsigned): A000032(n) (Lucas numbers). Alternating row sums: A099837(n+3).

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

%Y Cf. A049310, A053120, A108045, 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

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Last modified December 17 12:02 EST 2018. Contains 318200 sequences. (Running on oeis4.)