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A301417 Sums of positive coefficients in generalized Chebyshev polynomials of the first kind, for a family of 4 data.

%I A301417 #41 Mar 04 2024 00:31:13
%S A301417 1,4,19,98,516,2725,14400,76105,402229,2125864,11235643,59382770,
%T A301417 313850616,1658767513,8766940464,46335152161,244891172089,
%U A301417 1294302130684,6840663104371,36154365042098,191083538489436,1009917298758493,5337628549243344,28210506508524169
%N A301417 Sums of positive coefficients in generalized Chebyshev polynomials of the first kind, for a family of 4 data.
%C A301417 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 4 data, the sum of the positive coefficients in these polynomials is a(n). a(n+1)/a(n) approaches 1/(2^(1/4)-1). (For a family of 2 data, the coefficients of these polynomials give the Chebyshev polynomials of the first kind.)
%C A301417 More precisely, given a finite collection X:=(x(i), i =1..n) of data, the Girard-Waring formulae express the sum of the k-th powers of the data, S_k(X):=Sum(x(i)^k, i=1..n), in terms of the elementary symmetric polynomials in the data. The j-th elementary symmetric polynomial is s_j(X):=Sum(Product(x(i), x(i) in X_0), X_0 \subseteq X, where |X_0|=j). So the Girard-Waring formulae provide coefficients a(J,k) such that S_k(X)=Sum(a(J,k)*Product(s_j(X), j \in J), J:=(j(1), j(2), ...) where j(1)+j(2)+...=k). [Thus J is an integer partition of k.] By "mean powers" I mean T_k(X):=Sum(x(i)^k, i=1..n)/n. By the "mean symmetric polynomials" I mean t_j(X):=s_j(X)/binomial(n,j). The Girard-Waring mean formulae then provide coefficients b(J,k,n) such that T_k(X)=Sum(b(J,k,n)*Product(t_j(X), j in J), J:=(j(1), j(2), ...) where j(1)+j(2)+...=k). So the sums of positive coefficients that I reference, for a fixed data set size n, and a fixed power k, are Sum(b(J,k,n), J:=(j(1), j(2), ...) where j(1)+j(2)+...=k, such that b(J,k,n)>0).
%H A301417 Gregory Gerard Wojnar, <a href="/A301417/b301417.txt">Table of n, a(n) for n = 1..68</a>
%H A301417 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=4 p. 23.
%H A301417 Gregory Gerard Wojnar, <a href="/A301417/a301417.java.txt">Java program</a>.  Within the program, the variable I denotes the number of data; J denotes the exponent.
%H A301417 Michel Marcus, <a href="/A301417/a301417.gp.txt">pari script</a> (translated from java)
%H A301417 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5, 2, -2, -3, -1).
%F A301417 G.f.: (-x*(x+1)^3+1)/(x^5+3*x^4+2*x^3-2*x^2-5*x+1); this denominator equals (1-x)*(2-(1+x)^4).
%F A301417 a(n+5) = 5*a(n+4)+2*a(n+3)-2*a(n+2)-3*a(n+1)-a(n).
%t A301417 CoefficientList[Series[(-x (x + 1)^3 + 1)/(x^5 + 3 x^4 + 2 x^3 - 2 x^2 - 5 x + 1), {x, 0, 23}], x] (* _Michael De Vlieger_, Apr 07 2018 *)
%t A301417 LinearRecurrence[{5, 2, -2, -3, -1}, {1, 4, 19, 98, 516}, 24] (* _Jean-François Alcover_, Dec 02 2018 *)
%o A301417 (PARI) lista(4, nn) \\ use pari script link;  _Michel Marcus_, Apr 21 2018
%Y A301417 Cf. A302764, A024537, A195350, A301420, A301421, A301424.
%K A301417 nonn,easy
%O A301417 1,2
%A A301417 _Gregory Gerard Wojnar_, Mar 20 2018