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A008310
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Triangle of coefficients of Chebyshev polynomials T_n (x).
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13
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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; internal format)
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OFFSET
| 0,4
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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.
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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,
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.
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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 (mathar(AT)strw.leidenuniv.nl), Apr 20 2007
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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.
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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 (mathar(AT)strw.leidenuniv.nl), Apr 20 2007
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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
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CROSSREFS
| A039991 is essentially the same, 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.
Sequence in context: A198495 A084453 A097104 * A021431 A094936 A037892
Adjacent sequences: A008307 A008308 A008309 * A008311 A008312 A008313
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KEYWORD
| sign,tabf,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Additional comments and more terms from Henry Bottomley (se16(AT)btinternet.com), Dec 13 2000
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