login
Sums of positive coefficients in generalized Chebyshev polynomials of the first kind, for a family of 5 data.
6

%I #27 Apr 27 2018 11:03:28

%S 1,5,31,205,1376,9251,62210,418361,2813485,18920751,127242501,

%T 855708865,5754662616,38700243965,260260067876,1750255192001,

%U 11770508100345,79156948982921,532332378421395,3579947998967501,24075236064574376

%N Sums of positive coefficients in generalized Chebyshev polynomials of the first kind, for a family of 5 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 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.

%H Gregory Gerard Wojnar, <a href="/A301420/b301420.txt">Table of n, a(n) for n = 1..65</a>

%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=5 p. 23.

%F 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).

%F 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).

%o (PARI) lista(5, nn) \\ use pari script file in A301417; _Michel Marcus_, Apr 21 2018

%Y Cf. A302764, A024537, A195350, A301417, A301421, A301424.

%K nonn

%O 1,2

%A _Gregory Gerard Wojnar_, Mar 20 2018