%I #25 Apr 27 2018 11:03:08
%S 1,7,64,609,5846,56161,539540,5183417,49797685,478412117,4596160548,
%T 44155846113,424210322004,4075437640457,39153200900024,
%U 376149330687809,3613710136705565,34717331354145139,333533418773956668,3204294140706218329,30784024515164777522
%N Sums of positive coefficients of generalized Chebyshev polynomials of the first kind, for a family of 7 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 7 data, the sum of the positive coefficients in these polynomials is a(n). a(n+1)/a(n) approaches 1/(2^(1/7)-1). (For a family of 2 data, the coefficients of these polynomials give the Chebyshev polynomials of the first kind.) The sums of the negative coefficients are 1 less than the corresponding sums of the positive coefficients. See extended comment in A301417.
%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.
%F G.f.: (-x*(x+1)^6+1)/(x^2*(x^6+6*x^5+14*x^4+14*x^3-14*x-14)-8*x+1); this denominator equals (1-x)*(2-(1+x)^7) (conjectured).
%o (PARI) lista(7, nn) \\ use pari script file in A301417; _Michel Marcus_, Apr 21 2018
%Y Cf. A302764, A024537, A195350, A301417, A301420, A301421.
%K nonn
%O 1,2
%A _Gregory Gerard Wojnar_, Mar 20 2018