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A301420
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Sums of positive coefficients in generalized Chebyshev polynomials of the first kind, for a family of 5 data.
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6
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1, 5, 31, 205, 1376, 9251, 62210, 418361, 2813485, 18920751, 127242501, 855708865, 5754662616, 38700243965, 260260067876, 1750255192001, 11770508100345, 79156948982921, 532332378421395, 3579947998967501, 24075236064574376
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OFFSET
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1,2
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COMMENTS
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Re-express the Girard-Waring formulae to yield the mean powers in terms of the mean symmetric polynomials in the data values. Then for a family of 5 data, the sum of the positive coefficients in these polynomials is a(n). a(n+1)/a(n) approaches 1/(2^(1/5)-1). (For a family of 2 data, the coefficients of these polynomials give the Chebyshev polynomials of the first kind.) See extended comment in A301417.
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LINKS
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FORMULA
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G.f.: (-x*(x+1)^4+1)/(x^6+4*x^5+5*x^4-5*x^2-6*x+1); this denominator equals (1-x)*(2-(x+1)^5) (conjectured).
a(n+14) = 7*a(n+13) - a(n+12) - 6*a(n+11) + 2*a(n+10) - a(n+9) + 4*a(n+8) + a(n+7) + 4*a(n+5) + 2*a(n+4) - a(n+3) - 5*a(n+2) - 4*a(n+1) - a(n) (conjectured).
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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