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A008311
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Triangle of expansions of powers of x in terms of Chebyshev polynomials T_n (x).
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4
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1, 1, 1, 1, 3, 1, 3, 4, 1, 10, 5, 1, 10, 15, 6, 1, 35, 21, 7, 1, 35, 56, 28, 8, 1, 126, 84, 36, 9, 1, 126, 210, 120, 45, 10, 1, 462, 330, 165, 55, 11, 1, 462, 792, 495, 220, 66, 12, 1, 1716, 1287, 715, 286, 78, 13, 1, 1716, 3003, 2002, 1001, 364, 91, 14, 1, 6435, 5005, 3003
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| This triangle is the right half of Pascal's triangle (A007318), but with each number along the center of Pascal's triangle (except the 1 at the top) divided by 2. - Benjamin Schak (schak(AT)math.upenn.edu), Dec 02 2005
For n>=2 found in A002378, a(n)=A034869(n)/2, for all others a(n)=A034869(n). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 13 2006
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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.
H. J. Brothers, Pascal's Prism: Supplementary Material, http://www.brotherstechnology.com/docs/Pascal's_Prism_(supplement).pdf.
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LINKS
| 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].
Index entries for sequences related to Chebyshev polynomials.
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EXAMPLE
| Triangle begins:
1;
-, 1;
1, -, 1;
-, 3, -, 1;
3, -, 4, -, 1;
-, 10, -, 5, -, 1;
...
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MAPLE
| printf("1, ") ; for n from 1 to 20 do for j from n mod 2 to n by 2 do if j = 0 then printf("%d, ", binomial(n, (n-j)/2)/2) ; else printf("%d, ", binomial(n, (n-j)/2)) ; fi ; od ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 13 2006
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CROSSREFS
| With zeros: A100257.
Sequence in context: A093560 A173934 A131504 * A175721 A081772 A204217
Adjacent sequences: A008308 A008309 A008310 * A008312 A008313 A008314
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KEYWORD
| nonn,tabf,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Corrected by Philippe DELEHAM, Nov 12 2005
More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 13 2006
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