A008310 Triangle of coefficients of Chebyshev polynomials T_n(x).

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

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

Let T* denote the triangle obtained by replacing each number in this triangle by its absolute value. Then T* gives the coefficients for cos(nx) as a polynomial in cos x. - Clark Kimberling, Aug 04 2024

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.

Yaroslav Zolotaryuk, J. Chris Eilbeck, "Analytical approach to the Davydov-Scott theory with on-site potential", Physical Review B 63, p543402, Jan. 2001. The authors write, "Since the algebra of these is 'hyperbolic', contrary to the usual Chebyshev polynomials defined on the interval 0 <= x <= 1, we call the set of functions (21) the hyperbolic Chebyshev polynomials." (This refers to the triangle T* described in Comments.)

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].

Renato Ferreira Pinto Jr. and Nathaniel Harms, Testing Support Size More Efficiently Than Learning Histograms, arXiv:2410.18915 [cs.DS], 2024. See p. 40.

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), arXiv:math/9911076 [math.CO], 1999.

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

Wikipedia, Chebyshev polynomials.

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

From Paul Weisenhorn, Oct 02 2019: (Start)

T_n(x) = 2*x*T_(n-1)(x) - T_(n-2)(x), T_0(x) = 1, T_1(x) = x.

T_n(x) = ((x+sqrt(x^2-1))^n + (x-sqrt(x^2-1))^n)/2. (End)

From Peter Bala, Aug 15 2022: (Start)

T(n,x) = [z^n] ( z*x + sqrt(1 + z^2*(x^2 - 1)) )^n.

Sum_{k = 0..2*n} binomial(2*n,k)*T(k,x) = (2^n)*(1 + x)^n*T(n,x).

exp( Sum_{n >= 1} T(n,x)*t^n/n ) = Sum_{n >= 0} P(n,x)*t^n, where P(n,x) denotes the n-th Legendre polynomial. (End)

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
# second Maple program:
b:= proc(n) b(n):= `if`(n<2, 1, expand(2*b(n-1)-x*b(n-2))) end:
T:= n-> (p-> (d-> seq(coeff(p, x, d-i), i=0..d))(degree(p)))(b(n)):
seq(T(n), n=0..15);  # Alois P. Heinz, Sep 04 2019

Mathematica

Flatten[{1, Table[CoefficientList[ChebyshevT[n, x], x], {n, 1, 13}]}]//DeleteCases[#, 0, Infinity]& (* or *) Flatten[{1, Table[Table[((-1)^k*2^(n-2 k-1)*n*Binomial[n-k, k])/(n-k), {k, Floor[n/2], 0, -1}], {n, 1, 13}]}] (* Eugeniy Sokol, Sep 04 2019 *)

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 A350111

Adjacent sequences: A008307 A008308 A008309 * A008311 A008312 A008313

Keyword

sign,tabf,nice,easy,changed

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