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 A008310 Triangle of coefficients of Chebyshev polynomials T_n(x). 16
 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) Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the First Kind Wikipedia, 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) 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 A037892 Adjacent sequences:  A008307 A008308 A008309 * A008311 A008312 A008313 KEYWORD sign,tabf,nice,easy AUTHOR 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 December 12 15:11 EST 2019. Contains 329960 sequences. (Running on oeis4.)