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.)
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).
