login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A127674 Coefficient table for Chebyshev polynomials T(2*n,x) (increasing even powers x, without zeros). 11
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 (list; table; graph; refs; listen; history; text; internal format)
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. [From 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 [From Peter Bala, Jun 25 2009]

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

Wolfdieter Lang, Row polynomials.

Index entries for sequences related to Chebyshev polynomials.

FORMULA

a(n,m)=0 if n<m, a(0,0)=1 else 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

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).

Cf. A053120 (coefficients of T-polynomials, with interspersed zeros).

Cf. A086645, A063007, A108556. [Peter Bala, Oct 23 2008]

Cf. A063007.

Sequence in context: A015152 A021461 A075733 * A271316 A145901 A286724

Adjacent sequences:  A127671 A127672 A127673 * A127675 A127676 A127677

KEYWORD

sign,easy,tabl

AUTHOR

Wolfdieter Lang Mar 07 2007

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified June 21 15:14 EDT 2018. Contains 305622 sequences. (Running on oeis4.)