OFFSET
0,3
COMMENTS
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
LINKS
Bruno Berselli, Table of n, a(n) for n = 0..1000
G. G. Wojnar, D. S. Wojnar, and L. Q. Brin, Universal peculiar linear mean relationships in all polynomials, arXiv:1706.08381 [math.GM], 2017. See Table GW. n=3 p. 22.
Index entries for linear recurrences with constant coefficients, signature (4,0,-2,-1).
FORMULA
G.f.: (1-3*x-x^2)/((1-x)*(1-3*x-3*x^2-x^3)).
a(n) = 4*a(n-1) - 2*a(n-3) - a(n-4).
MAPLE
[seq(coeftayl((1-3*x-x^2)/(1-4*x+2*x^3+x^4), x = 0, k), k=0..25)]; # Muniru A Asiru, Mar 20 2018
MATHEMATICA
CoefficientList[Series[(1 - 3 x - x^2)/(1 - 4 x + 2 x^3 + x^4), {x, 0, 25}], x] (* Vincenzo Librandi, Mar 26 2013 *)
PROG
(PARI) Vec((1-3*x-x^2)/(1-4*x+2*x^3+x^4)+O(x^26))
(Magma) m:=26; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-3*x-x^2)/(1-4*x+2*x^3+x^4)));
(Maxima) makelist(coeff(taylor((1-3*x-x^2)/(1-4*x+2*x^3+x^4), x, 0, n), x, n), n, 0, 25);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Sep 16 2011
STATUS
approved