

A127672


Monic integer version of Chebyshev Tpolynomials (increasing powers).


82



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

0,1


COMMENTS

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 AbramowitzStegun 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(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
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_{doddpart(n)} C(2*n/d,x)
= Product_{doddpart(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, ..., n1; n>=1 (from those of the Chebyshev Tpolynomials).  Wolfdieter Lang, Sep 17 2011
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
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
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(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).
(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. 6974.  Wolfdieter Lang, Apr 28 2018
Some identities for the row polynomials R(n, x) following from the known ones for Chebyshev Tpolynomials (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(4x^2)), k >= 0, with the S row polynomials of A049310.
(4) R(2*k, x) = R(k, x^22), 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


REFERENCES

Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 6974.
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.


LINKS



FORMULA

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).
G.f. for mth column (signed triangle): 2/(1+x^2) if m=0 else (x^m)*(1x^2)/(1+x^2)^(m+1).
Riordan type matrix ((1x^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 = (2x*z)*S(x,z), with the o.g.f. S(x,z) = 1/(1  x*z + z^2) for the Spolynomials (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)/21,m1)/m.
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.
Chebyshev T(n,x) = Sum{m=0..n} a(n,m)*2^(m1)*x^m.  Wolfdieter Lang, Jun 03 2011
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
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.
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


EXAMPLE

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) = 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 Cpolynomials:
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


MAPLE

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


MATHEMATICA

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 *)


CROSSREFS

Row sums (signed): A057079(n1). Row sums (unsigned): A000032(n) (Lucas numbers). Alternating row sums: A099837(n+3).
Bisection: A127677 (even n triangle, without zero entries), ((1)^(nm))*A111125(n, m) (odd n triangle, without zero entries).


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STATUS

approved



