This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A127672 #137 Sep 19 2023 12:21:39
%S A127672 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 A127672 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 A127672 -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 A127672 Monic integer version of Chebyshev T-polynomials (increasing powers).
%C A127672 The row polynomials R(n,x) := Sum_{m=0..n} a(n,m)*x^m 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
%C A127672 This is a signed version of triangle A114525.
%C A127672 The unsigned column sequences (without zeros) are, for m=1..11: A005408, A000290, A000330, A002415, A005585, A040977, A050486, A053347, A054333, A054334, A057788.
%C A127672 The row polynomials R(n,x) := Sum_{m=0..n} a(n,m)*x*m, 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 A127672 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 A127672 R(n,x) = Product_{d|oddpart(n)} C(2*n/d,x)
%C A127672 = Product_{d|oddpart(n)} C(2^(k+1)*d,x),
%C A127672 with oddpart(n)=A000265(n), and 2^k is the largest power of 2 dividing n, where k=0,1,2,...
%C A127672 (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 A127672 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 A127672 The discriminants of the row polynomials R(n,x) are found under A193678. - _Wolfdieter Lang_, Aug 27 2011
%C A127672 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 A127672 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 A127672 From _Tom Copeland_, Nov 08 2015: (Start)
%C A127672 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 A127672 The partial sum of the first n+1 rows given t and y = t + 1/t is PS(n,t) = Sum_{k=0..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 A127672 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 A127672 (End)
%C A127672 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 A127672 From _Wolfdieter Lang_, May 05 2018: (Start)
%C A127672 Some identities for the row polynomials R(n, x) following from the known ones for Chebyshev T-polynomials (A053120) are:
%C A127672 (1) R(-n, x) = R(n, x).
%C A127672 (2) R(n*m, x) = R(n, R(m, x)) = R(m, R(n, x)).
%C A127672 (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 A127672 (4) R(2*k, x) = R(k, x^2-2), k >= 0.
%C A127672 (End)
%C A127672 For y = z^n + z^(-n) and x = z + z^(-1), Hirzebruch notes that y(z) = R(n,x) for the row polynomial of this entry. - _Tom Copeland_, Nov 09 2019
%D A127672 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 A127672 F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg 1994 pp. 77, 105.
%D A127672 R. Vein and P. Dale, Determinants and Their Applications in Mathematical Physics, Springer, 1999.
%H A127672 Robert Israel, <a href="/A127672/b127672.txt">Table of n, a(n) for n = 0..10010</a> (rows 0 to 140, flattened)
%H A127672 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 A127672 Tom 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 A127672 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 A127672 Gary Detlefs and Wolfdieter Lang, <a href="https://arxiv.org/abs/2304.12937">Improved Formula for the Multi-Section of the Linear Three-Term Recurrence Sequence</a>, arXiv:2304.12937 [math.CO], 2023.
%H A127672 Wolfdieter Lang, <a href="/A127672/a127672.pdf">Row polynomials.</a>
%H A127672 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 A127672 Wolfdieter Lang, <a href="https://arxiv.org/abs/2008.04300">On the Equivalence of Three Complete Cyclic Systems of Integers</a>, arXiv:2008.04300 [math.NT], 2020.
%H A127672 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A127672 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 A127672 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 A127672 Riordan type matrix ((1-x^2)/(1+x^2),x/(1+x^2)) if one puts a(0,0)=1 (instead of 2).
%F A127672 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 A127672 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 A127672 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 A127672 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 A127672 Chebyshev T(n,x) = Sum{m=0..n} a(n,m)*2^(m-1)*x^m. - _Wolfdieter Lang_, Jun 03 2011
%F A127672 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 A127672 From _Tom Copeland_, Nov 08 2015: (Start)
%F A127672 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 A127672 Examples: For n = 2, -2 + x^2 = sqrt(2 + 2 - 4*x^2 + x^4).
%F A127672 For n = 3, -3*x + x^3 = sqrt(2 - 2 + 9*x^2 - 6*x^4 + x^6).
%F A127672 (End)
%F A127672 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 + (-2*h2 + h1^2) x^2/2 + (-3*h1*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 A127672 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 A127672 Triangle begins:
%e A127672 n\m 0 1 2 3 4 5 6 7 8 9 10 ...
%e A127672 0: 2
%e A127672 1: 0 1
%e A127672 2: -2 0 1
%e A127672 3: 0 -3 0 1
%e A127672 4: 2 0 -4 0 1
%e A127672 5: 0 5 0 -5 0 1
%e A127672 6: -2 0 9 0 -6 0 1
%e A127672 7: 0 -7 0 14 0 -7 0 1
%e A127672 8: 2 0 -16 0 20 0 -8 0 1
%e A127672 9: 0 9 0 -30 0 27 0 -9 0 1
%e A127672 10: -2 0 25 0 -50 0 35 0 -10 0 1 ...
%e A127672 Factorization into minimal C-polynomials:
%e A127672 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 A127672 seq(seq(coeff(2*orthopoly[T](n,x/2),x,j),j=0..n),n=0..20); # _Robert Israel_, Aug 04 2015
%t A127672 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 A127672 Row sums (signed): A057079(n-1). Row sums (unsigned): A000032(n) (Lucas numbers). Alternating row sums: A099837(n+3).
%Y A127672 Bisection: A127677 (even n triangle, without zero entries), ((-1)^(n-m))*A111125(n, m) (odd n triangle, without zero entries).
%Y A127672 Cf. A049310, A053120, A108045, A263916.
%K A127672 sign,tabl,easy
%O A127672 0,1
%A A127672 _Wolfdieter Lang_, Mar 07 2007
%E A127672 Name changed and table rewritten by _Wolfdieter Lang_, Nov 08 2011