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A049310 Triangle of coefficients of Chebyshev's S(n,x) := U(n,x/2) polynomials (exponents in increasing order). 354
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). - From 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 NxN 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

REFERENCES

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; http://repository.wit.ie/1693/1/AoifeThesis.pdf

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.

P. Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, 70,1 (1997) 22-31.

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.

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. 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)

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

Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Index entries for sequences related to Chebyshev polynomials.

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)

EXAMPLE

The triangle T(n, k) begins

n\k  0  1   2   3   4   5   6    7   8   9  10  11 12 13 ...

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

12:  1  0 -21   0  70   0 -84    0  45   0 -11   0  1

13:  0  7   0 -56   0 126   0 -120   0  55   0 -12  0  1

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

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}] (* from 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.

Cf. 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.

Sequence in context: A234954 A180649 A191238 * A168561 A228783 A230425

Adjacent sequences:  A049307 A049308 A049309 * A049311 A049312 A049313

KEYWORD

easy,nice,sign,tabl,core

AUTHOR

Wolfdieter Lang

STATUS

approved

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Last modified April 20 05:05 EDT 2014. Contains 240779 sequences.