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A049310 Triangle of coefficients of Chebyshev's S(n,x) := U(n,x/2) polynomials (exponents in increasing order). 407
1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 1, 0, -3, 0, 1, 0, 3, 0, -4, 0, 1, -1, 0, 6, 0, -5, 0, 1, 0, -4, 0, 10, 0, -6, 0, 1, 1, 0, -10, 0, 15, 0, -7, 0, 1, 0, 5, 0, -20, 0, 21, 0, -8, 0, 1, -1, 0, 15, 0, -35, 0, 28, 0, -9, 0, 1, 0, -6, 0, 35, 0, -56, 0, 36, 0, -10, 0, 1, 1, 0, -21, 0, 70, 0, -84, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

G.f. for row polynomials S(n,x) (signed triangle): 1/(1-x*z+z^2). Unsigned triangle |a(n,m)| has Fibonacci polynomials F(n+1,x) as row polynomials with G.f. 1/(1-x*z-z^2). |a(n,m)| triangle has rows of Pascal's triangle A007318 in the even numbered diagonals (odd numbered ones have only 0's).

Row sums (unsigned triangle) A000045(n+1) (Fibonacci). Row sums (signed triangle) S(n,1) sequence = periodic(1,1,0,-1,-1,0) = A010892.

Alternating row sums A049347(n) = S(n,-1) = periodic(1,-1,0). - Wolfdieter Lang, Nov 04 2011

S(n,x) is the characteristic polynomial of the adjacency matrix of the n-path. - Michael Somos, Jun 24 2002

|T(n,k)|=number of compositions of n+1 into k+1 odd parts. Example: |T(7,3)|=10 because we have (1,1,3,3),(1,3,1,3),(1,3,3,1),(3,1,1,3),(3,1,3,1),(3,3,1,1), (1,1,1,5),(1,1,5,1),(1,5,1,1) and (5,1,1,1). - Emeric Deutsch, Apr 09 2005

S(n,x)= R(n,x) + S(n-2,x), n>=2, S(-1,x)=0, S(0,x)=1, R(n,x):=2*T(n,x/2) = sum(A127672(n,m)*x^m,m=0..n) (monic integer Chebyshev T-Polynomials). This is the rewritten so-called trace of the transfer matrix formula for the T-polynomials. - Wolfdieter Lang, Dec 02 2010

In a regular N-gon, inscribed in a unit circle, the side length is d(N,1)=2*sin(Pi/N). The length ratio R(N,k):=d(N,k)/d(N,1) for the (k-1)-th diagonal, with k from {2,3,...,floor(N/2)}, N>=4, equals S(k-1,x)= sin(k*Pi/N)/sin(Pi/N) with x=rho(N):=R(N,2)= 2*cos(Pi/N). Example: N=7 (heptagon),rho=R(7,2), sigma:=R(N,3)= S(2,rho)= rho^2-1. Motivated by the quoted paper by P. Steinbach. - Wolfdieter Lang, Dec 02 2010

From Wolfdieter Lang, Jul 12 2011: (Start)

In q- or basic analysis q-numbers are [n]_q :=S(n-1,q+1/q) = (q^n-(1/q)^n})/(q-1/q), with the row polynomials S(n,x), n>=0.

The zeros of the row polynomials S(n-1,x) are (from those of Chebyshev U-polynomials):

  x(n-1;k) = +/- t(k,rho(n)), k=1,...,ceiling((n-1)/2), n>=2, with t(n,x) the row polynomials of A127672 and rho(n):= 2*cos(Pi/n). The simple vanishing zero for even n appears here as +0 and -0.

Factorization of the row polynomials S(n-1,x), x>=1, in terms of the minimal polynomials of cos(2 pi/2), called Psi(n,x), with coefficients given by A181875/A181876:

  S(n-1,x) = (2^(n-1))*product(Psi(d,x/2), 2 < d | 2n), n>=1.

  (From the rewritten eq. (3) of the Watkins and Zeitlin reference, given under A181872.)

  (End)

The discriminants of the S(n,x) polynomials are found in A127670. [Wolfdieter Lang, Aug 03 2011]

This is an example for a subclass of Riordan convolution arrays (lower triangular matrices) called Bell arrays. See the L. W. Shapiro et al. reference under A007318. If a Riordan array is named (G(z),F(z)) with F(z)=z*Fhat(z), the o.g.f. for the row polynomials is G(z)/(1-x*z*Fhat(z)), and it becomes a Bell array if G(z)=Fhat(z). For the present Bell type triangle  G(z)=1/(1+z^2) (see the o.g.f. comment above). This leads to the o.g.f. for the column no. k, k>=0, x^k/(1+x^2)^(k+1) (see the formula section), the one for the row sums and for the alternating row sums (see comments above). The Riordan (Bell) A- and Z-sequences (defined in a W.Lang link under A006232, with references) have o.g.f.s 1-x*c(x^2) and -x*c(x^2), with the o.g.f. of the Catalan numbers A000108. Together they lead to a recurrence given in the formula section. - Wolfdieter Lang, Nov 04 2011

The determinant of the N x N matrix S(N,[x[1], ...., x[N]]) with elements S(m-1,x[n]), for n, m = 1, 2, ..., N, and for any x[n], is identical with the determinant of V(N,[x[1], ...., x[N]]) with elements x[n]^(m-1) (a Vandermondian, which equals product(x[j] - x[i], 1 <= i < j<= N)). This is a special instance of a theorem valid for any N >= 1 and any monic polynomial system p(m,x), m>=0 , with p(0,x) = 1. For this theorem see the Vein-Dale reference, p. 59. Thanks to L. Edson Jeffery for an email asking for a proof of the non-singularity of the matrix S(N,[x[1], ...., x[N]]) if and only if the x[j], j = 1..N, are pairwise different. - Wolfdieter Lang, Aug 26 2013

These S polynomials also appear in the context of modular forms. The rescaled Hecke operator T*_n = n^((1-k)/2)*T_n acting on modular forms of weight k satisfies T*_(p^n) = S(n, T*_p), for each prime p and positive integer n. See the Koecher-Krieg reference, p. 223. - Wolfdieter Lang, Jan 22 2016

For a shifted o.g.f. (mod signs), its compositional inverse, and connections to Motzkin and Fibonacci polynomials, non-crossing partitions and other combinatorial structures, see A097610. - Tom Copeland, Jan 23 2016

From M. Sinan Kul, Jan 30 2016: (Start)

Solutions of the Diophantine equation u^2 + v^2 - k*u*v = 1 for integer k given by (u(k,n), v(k,n)) = (S(n,k), S(n-1,k)) because of the Cassini-Simson identity: S(n,x)^2 - S(n+1,x)*S(n-1, x) = 1, after use of the S-recurrence. Note that S(-n, x) = -S(-n-2, x), n >= 1, and the periodicity of some S(n, k) sequences.

Hence another way to obtain the row polynomials would be to take powers of the matrix [x, -1; 1,0]: S(n, x) = ([x, -1; 1, 0])^n)[1,1], n >= 0.

See also a Feb 01 2016 comment on A115139 for a well known S(n, x) sum formula.

Then we have with the present T triangle

  A039834(n) = -i^(n+1)*T(n-1, k) where i is the imaginary unit and n >= 0.

  A051286(n) = Sum_{i=0..n}(T(n, i))^2 (see the

  Philippe Deléham, Nov 21 2005 formula),

  A181545(n) = Sum_{i=0..n+1}abs(T(n, i)^3),

  A181546(n) = Sum_{i=0..n+1}(T(n, i)^4,

  A181547(n) = Sum_{i=0..n+1}abs(T(n ,i)^5).

S(n ,0) = A056594(n), and for k = 1..10 the sequences S(n-1, k) with offset n = 0 are A128834, A001477, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189.

(End)

For more on the Diophantine equation presented by Kul, see the Ismail paper. - Tom Copeland, Jan 31 2016

The o.g.f. for the Legendre polynomials L(n,x) is 1 / sqrt(1- 2x*z + z^2), and squaring it gives the o.g.f. of U(n,x), A053117, so Sum_{k=0..n} L(k,x/2) L(n-k,x/2) = S(n,x). This gives S(n,x) = L(n/2,x/2)^2 + 2*Sum_{k=0...n/2-1} L(k,x/2) L(n-k,x/2) for n even and S(n,x) = 2*Sum_{k=0..(n-1)/2} L(k,x/2) L(n-k.x/2) for odd n. For a connection to elliptic curves and modular forms, see A053117. For the normalized Legendre polynomials, see A100258. For other properties and relations to other polynomials, see Allouche et al. - Tom Copeland, Feb 04 2016

LG(x,h1,h2) = -log(1 - h1*x + h2*x^2) = Sum_{n>0} F(n,-h1,h2,0,..,0) x^n/n is a log series generator of the bivariate row polynomials of A127672 with A127672(0,0) = 0 and where F(n,b1,b2,..,bn) are the Faber polynomials of A263916. Exp(LG(x,h1,h2)) = 1 / (1 - h1*x + h2*x^2 ) is the o.g.f. of the bivariate row polynomials of this entry. - Tom Copeland, Feb 15 2016

For distinct odd primes p and q the Legendre symbol can be written as Legendre(q,p) = Product_{k=1..P} S(q-1, 2*cos(2*Pi*k/p)), with P = (p-1)/2. See the Lemmermeyer reference, eq. (8.1) on p. 236. Using the zeros of S(q-1, x) (see above) one has S(q-1, x) = Product_{l=1..Q} (x^2 - (2*cos(Pi*l/q))^2), with Q = (q-1)/2. Thus  S(q-1, 2*cos(2*Pi*k/p)) = ((-4)^Q)*Product_{l=1..Q} (sin^2(2*Pi*k/p) - sin^2(Pi*l/q)) = ((-4)^Q)*Product_{m=1..Q} (sin^2(2*Pi*k/p) - sin^2(2*Pi*m/q)). For the proof of the last equality see a W. Lang comment on the triangle A057059 for n = Q and an obvious function f. This leads to Eisenstein's proof of the quadratic reciprocity law Legendre(q,p) = ((-1)^(P*Q)) * Legendre(p,q), See the Lemmermeyer reference, pp. 236-237.  - Wolfdieter Lang, Aug 28 2016

REFERENCES

Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, p. 223.

Franz Lemmermeyer, Reciprocity Laws. From Euler to Eisenstein, Springer, 2000.

D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, 1970; p. 232, Sect. 3.3.38.

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

R. Vein and P. Dale, Determinants and Their Applications in Mathematical Physics, Springer, 1999.

LINKS

T. D. Noe, Rows 0 to 100 of the triangle, flattened.

Wolfdieter Lang, First rows of the triangle.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], Table 22.8, p.797.

J. Allouche and G. Skordev, Schur congruences, Carlitz sequences of polynomials and automaticity, Discrete Mathematics, Vol. 214, Issue 1-3, 21 March 2000, p.21-49.

P. Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6

T. Copeland, Addendum to Elliptic Lie Triad

J. R. Dias, Properties and relationships of conjugated polyenes having a reciprocal eigenvalue spectrum - dendralene and radialene hydrocarbons , Croatica Chem. Acta, 77 (2004), 325-330. [p. 328].

S. R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008.

Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

M. Ismail,  One parameter generalizations of the Fibonacci and Lucas numbers, arXiv preprint arXiv:0606743v1 [math.CA], 2006.

W. Lang, Chebyshev S-polynomials: ten applications.

R. Sazdanovic, A categorification of the polynomial ring, slide presentation, 2011 (From Tom Copeland, Dec 27 2015)

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Index entries for sequences related to Chebyshev polynomials.

Index entries for "core" sequences

FORMULA

T(n, k) := 0 if n<k or n+k odd, else ((-1)^((n+k)/2+k))*binomial((n+k)/2, k); T(n, k) = -T(n-2, k)+T(n-1, k-1), T(n, -1) := 0 =: T(-1, k), T(0, 0)=1, T(n, k)= 0 if n<k or n+k odd; G.f. k-th column: (1 / (1 + x^2)^(k + 1)) * x^k. - Michael Somos, Jun 24 2002

T(n, k) = binomial((n+k)/2, (n-k)/2)*cos(Pi*(n-k)/2)*(1+(-1)^(n-k))/2. - Paul Barry, Aug 28 2005

Sum_{k=0..n} T(n, k)^2 = A051286(n). - Philippe Deléham, Nov 21 2005

Recurrence for the (unsigned) Fibonacci polynomials: F[1]=1, F[2]=x; for n>2, F[n] = x*F[n-1]+F[n-2].

From Wolfdieter Lang, Nov 04 2011: (Start)

The Riordan A- and Z-sequences, given in a comment above, lead together to the recurrence:

T(n,k) = 0 if n<k, if k=0 then T(0,0)=1 and

  T(n,0)=- sum(C(i)*T(n-1,2*i+1),i=0..floor((n-1)/2)), else T(n,k) = T(n-1,k-1) - sum(C(i)*T(n-1,k-1+2*i), i=1..floor((n-k)/2)), with the Catalan numbers C(n)=A000108(n).

(End)

The row polynomials satisfy also S(n,x) = 2*(T(n+2, x/2) - T(n, x/2))/(x^2-4) with the Chebyshev T-polynomials. Proof: Use the trace formula 2*T(n, x/2) = S(n, x) - S(n-2, x) (see the Dec 02 2010 comment above) and the S-recurrence several times. This is a formula which expresses the S- in terms of the T-polynomials. - Wolfdieter Lang, Aug 07 2014

From Tom Copeland, Dec 06 2015: (Start)

The non-vanishing, unsigned subdiagonals Diag_(2n) contain the elements D(n,k) = sum[j=0 to k, D(n-1,j)] = (k+1) (k+2) ... (k+n) / n! = binomial(n+k,n), so the o.g.f. for the subdiagonal is (1-x)^(-(n+1)). E.g., Diag_4 contains D(2,3) = D(1,0) + D(1,1) + D(1,2) + D(1,3) = 1 + 2 + 3 + 4 = 10 = binom(5,2). Diag_4 is shifted A000217; Diag_6, shifted A000292: Diag_8, shifted A000332; and Diag_10, A000389.

The non-vanishing anti-diagonals are signed rows of the Pascal triangle A007318.

For a reversed, unsigned version with the zeros removed, see A011973. (End)

EXAMPLE

The triangle T(n, k) begins

n\k  0  1   2   3   4   5   6    7   8   9  10  11

0:   1

1:   0  1

2:  -1  0   1

3:   0 -2   0   1

4:   1  0  -3   0   1

5:   0  3   0  -4   0   1

6:  -1  0   6   0  -5   0   1

7:   0 -4   0  10   0  -6   0    1

8:   1  0 -10   0  15   0  -7    0   1

9:   0  5   0 -20   0  21   0   -8   0   1

10: -1  0  15   0 -35   0  28    0  -9   0   1

11:  0 -6   0  35   0 -56   0   36   0 -10   0   1

... Reformatted and extended by Wolfdieter Lang, Oct 24 2012

For more rows see the link.

E.g., fourth row {0,-2,0,1} corresponds to polynomial S(3,x)= -2*x+x^3.

From Wolfdieter Lang, Jul 12 2011: (Start)

Zeros of S(3,x) with rho(4)= 2*cos(Pi/4) = sqrt(2):

  +/- t(1,sqrt(2)) = +/- sqrt(2) and

  +/- t(2,sqrt(2)) = +/- 0.

Factorization of S(3,x) in terms of Psi polynomials:

S(3,x) = (2^3)*Psi(4,x/2)*Psi(8,x/2) = x*(x^2-2).

(End)

From Wolfdieter Lang, Nov 04 2011: (Start)

A- and Z- sequence recurrence:

T(4,0) = - (C(0)*T(3,1) + C(1)*T(3,3)) = -(-2+1)=+1,

T(5,3) = -3 - 1*1 = -4.

(End)

MAPLE

A049310 := proc(n, k): binomial((n+k)/2, (n-k)/2)*cos(Pi*(n-k)/2)*(1+(-1)^(n-k))/2 end: seq(seq(A049310(n, k), k=0..n), n=0..11); # Johannes W. Meijer, Aug 08 2011

MATHEMATICA

t[n_, k_] /; EvenQ[n+k] = ((-1)^((n+k)/2+k))*Binomial[(n+k)/2, k]; t[n_, k_] /; OddQ[n+k] = 0; Flatten[Table[t[n, k], {n, 0, 12}, {k, 0, n}]][[;; 86]] (* Jean-François Alcover, Jul 05 2011 *)

Table[Coefficient[Fibonacci[n + 1, x], x, k], {n, 0, 10}, {k, 0, n}] (* Clark Kimberling, Aug 02 2011 *)

PROG

(PARI) {T(n, k) = if( k<0 || k>n || (n + k)%2, 0, (-1)^((n + k)/2 + k) * binomial((n + k)/2, k))} /* Michael Somos, Jun 24 2002 */

(Sage)

@CachedFunction

def A049310(n, k):

    if n< 0: return 0

    if n==0: return 1 if k == 0 else 0

    return A049310(n-1, k-1) - A049310(n-2, k)

for n in (0..9): [A049310(n, k) for k in (0..n)] # Peter Luschny, Nov 20 2012

CROSSREFS

Cf. A010892, A168561 (absolute values), A112552 (first column clipped).

Reflection of A053119. Without zeros: A053112.

Triangles of coefficients of Chebyshev's S(n,x+k) for k  = 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5 : A207824, A207823, A125662, A078812, A101950, A049310, A104562, A053122, A207815, A159764, A123967.

A053121 (inverse triangle).

Cf. A000217, A000292, A000332, A000389, A011973, A007318, A053117, A097610, A100258.

Sequence in context: A234954 A180649 A191238 * A168561 A253190 A228783

Adjacent sequences:  A049307 A049308 A049309 * A049311 A049312 A049313

KEYWORD

easy,nice,sign,tabl,core

AUTHOR

Wolfdieter Lang

EXTENSIONS

M. Sinan Kul's Jan 30 2016 comment edited by Wolfdieter Lang, Jan 31 2016 and Feb 01 2016

STATUS

approved

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Last modified September 30 04:15 EDT 2016. Contains 276618 sequences.