%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 Tpolynomials (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 AbramowitzStegun 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(N1,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^22 = phi1, 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),doddpart(n))
%C = product(C(2^{k+1}*d,x),doddpart(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, ..., n1; n>=1 (from those of the Chebyshev Tpolynomials).  _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(m1,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]^(m1). This is an instance of the general theorem given in the VeinDale 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)^(ks))*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(n1,t), and for t = e^(iq), y = 2 cos(q), and, therefore, a_n(2 cos(q)) = PS(n,e^(iq))  PS(n1,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. 6974.  _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 Tpolynomials (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(4x^2)), k >= 0, with the S row polynomials of A049310.
%C (4) R(2*k, x) = R(k, x^22), 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. 6974.
%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 T. Copeland, <a href="http://tcjpn.wordpress.com/2015/10/12/theellipticlietriadkdvandricatttequationsinfinigensandellipticgenera/">Addendum to Elliptic Lie Triad</a>
%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], 20112014.
%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 ngon</a>, arXiv:1210.1018 [math.GR], 20122017.
%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^(m1) with t(n,m):=A053120(n,m) (coefficients of Chebyshev Tpolynomials).
%F G.f. for mth column (signed triangle): 2/(1+x^2) if m=0 else (x^m)*(1x^2)/(1+x^2)^(m+1).
%F Riordan type matrix ((1x^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 )=(2x*z)*S(x,z), with the o.g.f. S(x,z)= 1/(1x*z+z^2) for the Spolynomials (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)/21,m1)/m.
%F Recursion for n >= 2 and m>=2: a(n,m) = a(n1,m1)a(n2,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^(m1))*x^m ).  _Wolfdieter Lang_, Jun 03 2011
%F R(n,x) = 2*T(n,x/2) = S(n,x)  S(n2,x), n>=0, with Chebyshev's T and Spolynomials, 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 halfangle 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^(m1)*x^m this becomes T(4,x)= 18*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 Cpolynomials:
%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(n1). Row sums (unsigned): A000032(n) (Lucas numbers). Alternating row sums: A099837(n+3).
%Y Bisection: A127677 (even n triangle, without zero entries), ((1)^(nm))*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
