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A127672
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Monic integer version of Chebyshev T-polynomials (increasing powers).
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82
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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, 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, -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
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OFFSET
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0,1
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COMMENTS
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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
This is a signed version of triangle A114525.
The unsigned column sequences (without zeros) are, for m=1..11: A005408, A000290, A000330, A002415, A005585, A040977, A050486, A053347, A054333, A054334, A057788.
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
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.
R(n,x) = Product_{d|oddpart(n)} C(2*n/d,x)
= Product_{d|oddpart(n)} C(2^(k+1)*d,x),
with oddpart(n)=A000265(n), and 2^k is the largest power of 2 dividing n, where k=0,1,2,...
(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]
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
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
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
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.
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.)
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).
(End)
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
Some identities for the row polynomials R(n, x) following from the known ones for Chebyshev T-polynomials (A053120) are:
(1) R(-n, x) = R(n, x).
(2) R(n*m, x) = R(n, R(m, x)) = R(m, R(n, x)).
(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.
(4) R(2*k, x) = R(k, x^2-2), k >= 0.
(End)
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
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REFERENCES
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Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.
F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg 1994 pp. 77, 105.
R. Vein and P. Dale, Determinants and Their Applications in Mathematical Physics, Springer, 1999.
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LINKS
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FORMULA
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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).
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).
Riordan type matrix ((1-x^2)/(1+x^2),x/(1+x^2)) if one puts a(0,0)=1 (instead of 2).
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).
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
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.
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.
Chebyshev T(n,x) = Sum{m=0..n} a(n,m)*2^(m-1)*x^m. - Wolfdieter Lang, Jun 03 2011
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
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.
Examples: For n = 2, -2 + x^2 = sqrt(2 + 2 - 4*x^2 + x^4).
For n = 3, -3*x + x^3 = sqrt(2 - 2 + 9*x^2 - 6*x^4 + x^6).
(End)
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
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EXAMPLE
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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.
Triangle begins:
n\m 0 1 2 3 4 5 6 7 8 9 10 ...
0: 2
1: 0 1
2: -2 0 1
3: 0 -3 0 1
4: 2 0 -4 0 1
5: 0 5 0 -5 0 1
6: -2 0 9 0 -6 0 1
7: 0 -7 0 14 0 -7 0 1
8: 2 0 -16 0 20 0 -8 0 1
9: 0 9 0 -30 0 27 0 -9 0 1
10: -2 0 25 0 -50 0 35 0 -10 0 1 ...
Factorization into minimal C-polynomials:
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
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MAPLE
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seq(seq(coeff(2*orthopoly[T](n, x/2), x, j), j=0..n), n=0..20); # Robert Israel, Aug 04 2015
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MATHEMATICA
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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 *)
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CROSSREFS
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Row sums (signed): A057079(n-1). Row sums (unsigned): A000032(n) (Lucas numbers). Alternating row sums: A099837(n+3).
Bisection: A127677 (even n triangle, without zero entries), ((-1)^(n-m))*A111125(n, m) (odd n triangle, without zero entries).
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KEYWORD
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EXTENSIONS
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STATUS
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approved
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