cp's OEIS Frontend

Rewrite the Girard-Waring formulae to express the mean powers in terms of the mean symmetric functions of the data values; the results are polynomials in the mean symmetric polynomials, indexed by the power n. Then for 3 data points, the sum of the positive coefficients in the n-th such polynomial is a(n). a(n+1)/a(n) approaches 1/(2^(1/3)-1). See extended comment in A301417. - Gregory Gerard Wojnar, Mar 19 2018

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).

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.

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.

Number the rows of the triangle beginning with n=0. For each row construct a degree n polynomial with regularly decreasing powers, denoting the polynomial as f_n(x); e.g., for row 2 we have f_2(x)=1x^2+2x+0. Then construct g_n(x)=x^2*f_{n-1}(x)-(n+1)x+1. It obtains that g_n(x)=(1-x)(2-(1+x)^n). These g_n(x) are the denominators of the generating functions for the following sequences: A024537 (n=2); A195350 (n=3); A301417 (n=4); A301420 (n=5); A301421 (n=6); A301424 (n=7). For these sequences the asymptotic term-to-term ratios are 1/(2^(1/n)-1). The numerators of the generating functions are 1-x(x+1)^(n-1).