A127674 - OEIS

A127674

Coefficient table for Chebyshev polynomials T(2*n,x) (increasing even powers x, without zeros).

1, -1, 2, 1, -8, 8, -1, 18, -48, 32, 1, -32, 160, -256, 128, -1, 50, -400, 1120, -1280, 512, 1, -72, 840, -3584, 6912, -6144, 2048, -1, 98, -1568, 9408, -26880, 39424, -28672, 8192, 1, -128, 2688, -21504, 84480, -180224, 212992, -131072, 32768, -1, 162, -4320, 44352, -228096, 658944, -1118208

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OFFSET

0,3

COMMENTS

Let C_n be the root lattice generated as a monoid by {+-2*e_i: 1 <= i <= n; +-e_i +- e_j: 1 <= i not equal to j <= n}. Let P(C_n) be the polytope formed by the convex hull of this generating set. Then the rows of (the signless version of) this array are the f-vectors of a unimodular triangulation of P(C_n) [Ardila et al.]. See A086645 for the corresponding array of h-vectors for these type C_n polytopes. See A063007 for the array of f-vectors for type A_n polytopes and A108556 for the array of f-vectors associated with type D_n polytopes. - Peter Bala, Oct 23 2008

REFERENCES

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990. p. 37, eq.(1.96) and p. 4. eq.(1.10).

LINKS

Table of n, a(n) for n=0..51.

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].

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 795.

F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes and growth series of root lattices, arXiv:0809.5123 [math.CO], 2008.

G. Boros, V. H. Moll, Landen transformations and the integration of rational functions, Math. Comp. 71 (238) (2001) 649-668, absolute values in Lemma A.3.

C. Lanczos, Applied Analysis (Annotated scans of selected pages) See p. 516.

Wolfdieter Lang, Row polynomials.

Richard J. Mathar, Chebyshev approximation of x^m (-log x)^l in the interval 0 <= x <= 1, arXiv:2408.15212 [math.CA], 2024. See p. 1.

Index entries for sequences related to Chebyshev polynomials.

FORMULA

a(n,m) = 0 if n < m, a(0,0) = 1; otherwise a(n,m) = ((-1)^(n-m))*(2^(2*m-1))*binomial(n+m,2*m)*(2*n)/(n+m).

O.g.f.: (1 + z*(1 - 2*x))/((1 + z)^2 - 4*x*z) = 1 + (-1 + 2*x)*z + (1 - 8*x + 8*x^2)*z^2 + ... . [Peter Bala, Oct 23 2008] For the t-polynomials actually with x -> x^2. - Wolfdieter Lang, Aug 02 2014

Denoting the row polynomials by R(n,x) we have exp( Sum_{n >= 1} R(n,x)*z^n/n ) = 1/sqrt( (1 + z)^2 - 4*x*z ) = 1 + (-1 + 2*x)*z + (1 - 6*x + 6*x^2)*z^2 + ..., the o.g.f. for a signed version of A063007. - Peter Bala, Sep 02 2015

The n-th row polynomial equals T(n, 2*x - 1). - Peter Bala, Jul 09 2023

EXAMPLE

[1];

[-1,2];

[1,-8,8];

[-1,18,-48,32];

[1,-32,160,-256,128];

...

See a link for the row polynomials.

The T-polynomial for row n=3, [-1,18,-48,32], is T(2*3,x) = -1*x^0 + 18*x^2 - 48*x^4 + 32*x^6.

CROSSREFS

Cf. A075733 (different signs and offset). A084930 (coefficients of odd-indexed T-polynomials).