%I #130 Oct 27 2024 11:46:52

%S 1,1,-1,2,-3,4,1,-8,8,5,-20,16,-1,18,-48,32,-7,56,-112,64,1,-32,160,

%T -256,128,9,-120,432,-576,256,-1,50,-400,1120,-1280,512,-11,220,-1232,

%U 2816,-2816,1024,1,-72,840,-3584,6912,-6144,2048,13,-364,2912,-9984,16640,-13312,4096

%N Triangle of coefficients of Chebyshev polynomials T_n(x).

%C 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

%C 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

%D 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.

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

%D 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.)

%H R. J. Mathar, <a href="/A008310/b008310.txt">Table of n, a(n) for n = 0..2600</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Renato Ferreira Pinto Jr. and Nathaniel Harms, <a href="https://arxiv.org/abs/2410.18915">Testing Support Size More Efficiently Than Learning Histograms</a>, arXiv:2410.18915 [cs.DS], 2024. See p. 40.

%H D. Foata and G.-N. Han, <a href="https://irma.math.unistra.fr/~foata/paper/pub71.html">Nombres de Fibonacci et polynomes orthogonaux</a>

%H C. Lanczos, <a href="/A002457/a002457.pdf">Applied Analysis</a> (Annotated scans of selected pages)

%H I. Rivin, <a href="https://arxiv.org/abs/math/9911076">Growth in free groups (and other stories)</a>, arXiv:math/9911076 [math.CO], 1999.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html">Chebyshev Polynomial of the First Kind</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.

%F 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

%F From _Paul Weisenhorn_, Oct 02 2019: (Start)

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

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

%F From _Peter Bala_, Aug 15 2022: (Start)

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

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

%F 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)

%e 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.

%e From _Wolfdieter Lang_, Aug 02 2014: (Start)

%e This irregular triangle a(n,k) begins:

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

%e 0: 1

%e 1: 1

%e 2: -1 2

%e 3: -3 4

%e 4: 1 -8 8

%e 5: 5 -20 16

%e 6: -1 18 -48 32

%e 7: -7 56 -112 64

%e 8: 1 -32 160 -256 128

%e 9: 9 -120 432 -576 256

%e 10: -1 50 -400 1120 -1280 512

%e 11: -11 220 -1232 2816 -2816 1024

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

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

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

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

%e ...

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

%e (End)

%p 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

%p # second Maple program:

%p b:= proc(n) b(n):= `if`(n<2, 1, expand(2*b(n-1)-x*b(n-2))) end:

%p T:= n-> (p-> (d-> seq(coeff(p, x, d-i), i=0..d))(degree(p)))(b(n)):

%p seq(T(n), n=0..15); # _Alois P. Heinz_, Sep 04 2019

%t 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 *)

%Y 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.

%Y Reflection of A028297. Cf. A008312, A053112.

%Y 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.

%Y Cf. A053120.

%K sign,tabf,nice,easy

%O 0,4

%A _N. J. A. Sloane_

%E Additional comments and more terms from _Henry Bottomley_, Dec 13 2000

%E Edited: Corrected Cf. A039991 statement. Cf. A053120 added. - _Wolfdieter Lang_, Aug 06 2014