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A008310 Triangle of coefficients of Chebyshev polynomials T_n (x). 14
1, 1, -1, 2, -3, 4, 1, -8, 8, 5, -20, 16, -1, 18, -48, 32, -7, 56, -112, 64, 1, -32, 160, -256, 128, 9, -120, 432, -576, 256, -1, 50, -400, 1120, -1280, 512, -11, 220, -1232, 2816, -2816, 1024, 1, -72, 840, -3584, 6912, -6144, 2048, 13, -364, 2912, -9984, 16640, -13312, 4096 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

The row length sequence of this irregular array is A008619(n), n >= 0. Even or odd powers appear in increasing order starting with 1 or x for even or odd row numbers n, respectively. This is the standard triangle A053120 with 0 deleted. - Wolfdieter Lang, Aug 02 2014

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.

E. A. Guilleman, Synthesis of Passive Networks, Wiley, 1957, p. 593.

LINKS

R. J. Mathar, Table of n, a(n) for n = 0..2600

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].

D. Foata and G.-N. Han, Nombres de Fibonacci et polynomes orthogonaux

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

I. Rivin, Growth in free groups (and other stories)

Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the First Kind

Index entries for sequences related to Chebyshev polynomials.

FORMULA

a(n,m)=2^(m-1)*n*(-1)^[(n-m)/2]*[(n+m)/2-1]!/{[(n-m)/2]! m!} if n>0. - R. J. Mathar, Apr 20 2007

EXAMPLE

Rows are: (1), (1), (-1,2), (-3,4), (1,-8,8), (5,-20,16) etc., since if c = cos(x): cos(0x) = 1, cos(1x) = 1c; cos(2x) = -1+2c^2; cos(3x) = -3c+4c^3, cos(4x) = 1-8c^2+8c^4, cos(5x) = 5c-20c^3+16c^5, etc.

From Wolfdieter Lang, Aug 02 2014: (Start)

This irregular triangle a(n,k) begins:

n\k   0    1     2      3      4      5      6      7 ...

0:    1

1:    1

2:   -1    2

3:   -3    4

4:    1   -8     8

5:    5  -20    16

6:   -1   18   -48     32

7:   -7   56  -112     64

8:    1  -32   160   -256    128

9:    9 -120   432   -576    256

10:  -1   50  -400   1120  -1280    512

11: -11  220 -1232   2816  -2816   1024

12:   1  -72   840  -3584   6912  -6144   2048

13:  13 -364  2912  -9984  16640 -13312   4096

14:  -1   98 -1568   9408 -26880  39424 -28672   8192

15: -15  560 -6048  28800 -70400  92160 -61440  16384

...

T(4,x) = 1 - 8*x^2 + 8*x^4, T(5,x) = 5*x - 20*x^3 +16*x^5.

(End)

MAPLE

A008310 := proc(n, m) local x ; coeftayl(simplify(ChebyshevT(n, x), 'ChebyshevT'), x=0, m) ; end: i := 0 : for n from 0 to 100 do for m from n mod 2 to n by 2 do printf("%d %d ", i, A008310(n, m)) ; i := i+1 ; od ; od ; # R. J. Mathar, Apr 20 2007

MATHEMATICA

Table[PolynomialMod[ChebyshevT[2 k+1, x]-1, ChebyshevU[k, x]+ChebyshevU[k-1, x]], {k, 10}] (* Takashi Tokita (butaneko(AT)fa2.so-net.ne.jp), Aug 19 2005 *)

CROSSREFS

A039991 is a row reversed version, but has zeros which enable the triangle to be seen. Columns/diagonals are A011782, A001792, A001793, A001794, A006974, A006975, A006976 etc.

Reflection of A028297. Cf. A008312, A053112.

Row sums are one. Polynomial evaluations include A001075 (x=2), A001541 (x=3), A001091, A001079, A023038, A011943, A001081, A023039, A001085, A077422, A077424, A097308, A097310, A068203.

Cf. A053120.

Sequence in context: A198495 A084453 A097104 * A021431 A094936 A037892

Adjacent sequences:  A008307 A008308 A008309 * A008311 A008312 A008313

KEYWORD

sign,tabf,nice,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Additional comments and more terms from Henry Bottomley, Dec 13 2000

Edited: Corrected Cf. A039991 statement. Cf. A053120 added. - Wolfdieter Lang, Aug 06 2014

STATUS

approved

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Last modified February 23 04:57 EST 2018. Contains 299473 sequences. (Running on oeis4.)