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A049310 Triangle of coefficients of Chebyshev's S(n,x) := U(n,x/2) polynomials (exponents in increasing order). 373
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

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.

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)

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.

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

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.

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

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 October 25 17:33 EDT 2014. Contains 248557 sequences.