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Search: id:A127673
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| A127673 |
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One half of even powers of 2*x in terms of Chebyshev's T-polynomials. |
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3
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| 1, 1, 1, 1, 4, 3, 1, 6, 15, 10, 1, 8, 28, 56, 35, 1, 10, 45, 120, 210, 126, 1, 12, 66, 220, 495, 792, 462, 1, 14, 91, 364, 1001, 2002, 3003, 1716, 1, 16, 120, 560, 1820, 4368, 8008, 11440, 6435, 1, 18, 153, 816, 3060, 8568, 18564, 31824, 43758, 24310, 1, 20, 190
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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See A122366 for one half of odd powers of 2*x in terms of Chebyshev's T-polynomials.
This is, for n>=1, the left half of Pascal's triangle for even rows with the central coefficients divided by 2.
The signed version of this triangle, b(n,k):=a(n,k)*(-1)^(n-k), appears in the formula (1/2)*(2*sin(phi))^(2*n) = sum(b(n,k)*cos(2*(n-k)*phi),k=0..n-1)+a(n,n).
Correspondingly, (1/2)*(4*(1-x^2))^n = sum(b(n,k)*T(2*(n-k),x),k=0..n-1)+a(n,n).
The proofs follow from Euler's formula 2*x=2*cos(phi)=exp(i*phi)+exp(-i*phi) or 2*sqrt(1-x^2)=2*sin(phi)=(exp(i*phi)-exp(-i*phi))/i and the binomial formula.
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REFERENCES
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Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990. p. 54-5, Ex.1.5.31.
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LINKS
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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].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 795.
Index entries for sequences related to Chebyshev polynomials.
Index entries for triangles and arrays related to Pascal's triangle
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FORMULA
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a(n,k)=biniomial(2*n,k), k=0,...,n-1 and a(n,n)=binomial(2*n,n)/2, n>=1. Instead of a(0,0)=1 one should take 1/2.
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EXAMPLE
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[1/2];[1,1];[1,4,3];[1,6,15,10];[1,8,28,56,35];...
Row n=3: [1,6,15,20/2=10] appears in ((2*x)^6)/2 = 1*T(6,x) + 6*T(4,x) + 15*T(2,x) + 10.
Row n=3: [1,6,15,20/2=10] appears in ((2*cos(phi))^6)/2 = 1*cos(6*phi) + 6*cos(4*phi) + 15*cos(2*phi) + 10.
The signed row n=3, [ -1,6,-15,+20/2=10], appears in ((4*(1-x^2))^3)/2 = -1*T(6,x) + 6*T(4,x) - 15*T(2,x) + 10).
The signed row n=3, [ -1,6,-15,+20/2=10], appears in ((2*sin(phi))^6)/2 = -1*cos(6*phi) + 6*cos(4*phi) - 15*cos(2*phi) + 10.
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CROSSREFS
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Cf. A122366, A008314, A008311.
Sequence in context: A020839 A087274 A103552 this_sequence A016698 A038763 A128007
Adjacent sequences: A127670 A127671 A127672 this_sequence A127674 A127675 A127676
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KEYWORD
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nonn,tabl,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Mar 07 2007
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