|
|
A301424
|
|
Sums of positive coefficients of generalized Chebyshev polynomials of the first kind, for a family of 7 data.
|
|
5
|
|
|
1, 7, 64, 609, 5846, 56161, 539540, 5183417, 49797685, 478412117, 4596160548, 44155846113, 424210322004, 4075437640457, 39153200900024, 376149330687809, 3613710136705565, 34717331354145139, 333533418773956668, 3204294140706218329, 30784024515164777522
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
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 7 data, the sum of the positive coefficients in these polynomials is a(n). a(n+1)/a(n) approaches 1/(2^(1/7)-1). (For a family of 2 data, the coefficients of these polynomials give the Chebyshev polynomials of the first kind.) The sums of the negative coefficients are 1 less than the corresponding sums of the positive coefficients. See extended comment in A301417.
|
|
LINKS
|
|
|
FORMULA
|
G.f.: (-x*(x+1)^6+1)/(x^2*(x^6+6*x^5+14*x^4+14*x^3-14*x-14)-8*x+1); this denominator equals (1-x)*(2-(1+x)^7) (conjectured).
|
|
PROG
|
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|