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Sums of positive coefficients of generalized Chebyshev polynomials of the first kind, for a family of 6 data.
6

%I #30 Aug 18 2021 15:27:07

%S 1,6,46,371,3026,24707,201748,1647429,13452565,109850886,897019828,

%T 7324880157,59813470848,488424550081,3988374821616,32568251770049,

%U 265945672309613,2171657880797162,17733313387923690,144806604435722311,1182461068019218530,9655734852907204771

%N Sums of positive coefficients of generalized Chebyshev polynomials of the first kind, for a family of 6 data.

%C 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 6 data, the sum of the positive coefficients in these polynomials is a(n). a(n+1)/a(n) approaches 1/(2^(1/6)-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.

%H Gregory Gerard Wojnar, <a href="/A301421/b301421.txt">Table of n, a(n) for n = 1..62</a> [a(21) corrected by _Georg Fischer_, Aug 18 2021]

%H G. G. Wojnar, D. S. Wojnar, and L. Q. Brin, <a href="http://arxiv.org/abs/1706.08381">Universal peculiar linear mean relationships in all polynomials</a>, arXiv:1706.08381 [math.GM], 2017. See Table GW.n=6 p. 24.

%F G.f.: (-x*(x+1)^5+1)/(x^7+5*x^6+9*x^5+5*x^4-5*x^3-9*x^2-7*x+1); this denominator equals (1-x)*(2-(1+x)^6) (conjectured).

%o (PARI) lista(6, nn) \\ use pari script file in A301417; _Michel Marcus_, Apr 21 2018

%Y Cf. A301764, A024537, A195350, A301417, A301420, A301424.

%K nonn

%O 1,2

%A _Gregory Gerard Wojnar_, Mar 20 2018

%E a(21) corrected by _Georg Fischer_, Aug 18 2021